foundations of geometry - math with mr....

Foundations ofGeometryFoundations ofGeometry

374 Chapter 8

KEYWORD: MT8CA Ch8

8A Plane Geometry8-1 Points, Lines, Planes,

and Angles

LAB Construct Bisectors andCongruent Angles

8-2 Geometric Relationships

8-3 Angle Relationships

8-4 Triangles

LAB Identify and ConstructAltitudes

8-5 Coordinate Geometry

LAB Angles in Polygons

8B Congruence andTransformations

8-6 Congruent Polygons

8-7 Transformations

LAB Combine Transformations

8-8 Tessellations

Devils PostpileNational Monument,Mammoth Lakes

Geometric figures can berepresented by real-worldobjects, such as thehexagonal columns ofDevils Postpile.

Foundations of Geometry 375

VocabularyChoose the best term from the list to complete each sentence.

1. In the __?__ (4, �3), 4 is the __?__, and �3 is the __?__.

2. The __?__ and the __?__ divide the __?__ into four sections.

3. The point (0, 0) is called the __?__.

4. The point (0, �3) lies on the __?__, while the point (�2, 0) lies on the __?__.

Complete these exercises to review skills you will need for this chapter.

Ordered Pairs Write the coordinates of the indicated points.

5. point A 6. point B

7. point C 8. point D

9. point E 10. point F

11. point G 12. point H

Similar FiguresTell whether the figures in each pair appear to be similar.

13. 14.

Solve EquationsSolve each equation.

15. 2p � 18 16. 7 � h � 21 17. �3x

� � 9 18. y � 6 � 16

19. 4d � 1 � 13 20. �2q � 3 � 3 21. 4(z � 1) � 16 22. x � 3 � 4x � 23

2�2�4

coordinate plane

ordered pair

origin

x-axis

x-coordinate

y-axis

y-coordinate

The information below “unpacks” the standards. The Academic Vocabulary ishighlighted and defined to help you understand the language of the standards.Refer to the lessons listed after each standard for help with the math terms andphrases. The Chapter Concept shows how the standard is applied in this chapter.

California Standard

Chapter Concept

Academic Vocabulary

Standards AF3.3 and MG3.3 are also covered in this chapter. To see these standards unpacked, go to Chapter 4, p. 166 (MG3.3)and Chapter 7, p. 320 (AF3.3).

MG3.1 Identify and basic of geometric figures(e.g., altitudes, midpoints, diagonals,angle bisectors, and perpendicularbisectors; central angles, radii,diameters, and chords of circles) byusing a compass and straightedge.

(Labs 8-1, 8-5)

elementsconstruct construct make or draw using tools

elements parts of

You learn parts of geometricfigures such as points, lines,angles, and planes.

MG3.2 Understand and usecoordinate graphs to simplefigures, determine lengths and areasrelated to them, and determine their

under translations andreflections.

(Lessons 8-5, 8-7; Lab 8-7)

plotplot draw on a graph

image a picture

You draw geometric figures ona graph and then draw apicture of the figure after it ismoved.

MG3.4 Demonstrate anunderstanding of conditions that

two geometrical figuresare congruent and whatcongruence means about therelationship between the sides andangles of the two figures.

(Lesson 8-6)

indicate

indicate show You learn how to show that twofigures are congruent.

376 Chapter 8

MG3.6 Identify elements ofthree-dimensional geometric objects(e.g., diagonals of rectangular solids)and describe how two or moreobjects are related in (e.g.,skew lines, the possible waysthree planes might .)

(Lesson 8-2)

intersect

space a three-dimensional region

intersect meet or cross

You learn how geometricfigures in a three-dimensionalregion may meet.

Foundations of Geometry 377

Writing Strategy: Use Your Own WordsExplaining a concept in your own words will help you better understand it.For example, learning to solve two-step inequalities might seem difficult ifthe textbook does not use the same words that you would use.

As you work through each lesson, do the following:

• Identify the important concepts.

• Use your own words to explain the concepts.

• Use examples to help clarify your thoughts.

Rewrite each statement in your own words.

1. Like terms can be grouped together because they have the samevariable raised to the same power.

2. If an equation contains fractions, consider multiplying both sides of theequation by the least common denominator (LCD) to clear the fractionsbefore you isolate the variable.

3. To solve multi-step equations with variables on both sides, firstcombine like terms and then clear fractions. Then add or subtractvariable terms on both sides so that the variable occurs on only one sideof the equation. Then use properties of equality to isolate the variable.

Try This

Solving a two-stepinequality uses the sameinverse operations as solvinga two-step equation.

Multiplying or dividing an inequality by a negativenumber reverses theinequality symbol.

What Miguel Reads What Miguel Writes

Solve a two-step inequality like a two-step equation. Use operations that undo each other.

When you multiply or divide bya negative number, switch theinequality symbol so that itfaces the opposite direction.

�4y 8 Divide by �4 andy �2 switch the symbol.�

CaliforniaStandards

English-Language Arts, Reading 1.3

CaliforniaStandards

Preparation for MG3.1 Identifyand construct basic elements ofgeometric figures (e.g., altitudes,midpoints, diagonals, anglebisectors, and perpendicularbisectors; central angles, radii,diameters, and chords of circles)by using a compass andstraightedge.

Who uses this? Mapmakers use representations ofpoints and lines to create maps. (See Exercises 28–32.)

Vocabularypointlineplanesegmentrayangleacute angleright angleobtuse anglestraight anglecomplementary anglessupplementary angles

Points, lines, and planes are the building blocks of geometry.Segments, rays, and angles are defined in terms of these basic figures.

A names a location. • A point A

A is perfectly straight and extends forever in both directions.

A is a perfectly flat surface thatextends forever in all directions.

A , or line segment, is the part of a line between two points.

A is part of a line that starts at one point and extends forever in one direction.

segment

BC�� is read as “line BC.” GH�� is read as “segment GH.” KJ�� is read as “ray KJ.”

Naming Lines, Planes, Segments, and Rays

Use the diagram to name each figure.a line

Possible answers: Any 2 points on the line can QS��, QR��, or RS�� be used.

a plane

Possible answers: Any 3 points in the plane that plane Z or plane QRT form a triangle can name a plane.

four segments Write the 2 points in any order,

Possible answers: QR��, RS��, RT��, QS�� for example, Q�R� or R�Q�.

five rays

RQ��, RS��, RT��, SQ��, QS�� Write the endpoint first.

E X A M P L E 1

line �, or BC

EPplane P, orplane DEFF

378 Chapter 8 Foundations of Geometry

8-1 Points, Lines, Planes,and Angles

When naming a ray,always write theendpoint first.

Classifying Angles

Use the diagram to name each figure.

a right angle two acute angles

�DEC �AED, �CEB

two obtuse angles

�AEC, �DEB m�AEC � 150°; m�DEB � 120°

a pair of complementary angles

�AED, �CEB m�AED � m�CEB � 60° � 30° � 90°

two pairs of supplementary angles

�AED, �DEB m�AED � m�DEB � 60° � 120° � 180°�AEC, �CEB m�AEC � m�CEB � 150° � 30° � 180°

An (�) is formed by two rays, or sides, with a commonendpoint called the vertex. You can name an angle several ways:by its vertex, by its vertex and a point on each ray, or by a number.When three points are used, the middle point must be the vertex.

Angles are usually measured in degrees (°). Since there are 360° in a circle, one degree is �3

160� of a circle.

30°90°

m�AEC is read as“the measure ofangle AEC.”

2E X A M P L E

8-1 Points, Lines, Planes, and Angles 379

�Y, �XYZ,�ZYX, or �1

Think and Discuss

1. Explain how XY�� is different from XY��.

2. Name all the possible lines, segments, and rays that includepoints J and K.

65 115o o

The measure of anis greater

than 0° and less than 90°.

The measure of a

is 180°.

The measure of a

is 90°.

are two angles whose

measures add to 90°.

The measure of an

is greater

than 90° and less than 180°.

are two angles whose

measures add to 180°.

Supplementary angles

obtuse angle

Complementary angles

right angle

straight angle

acute angle

8-1 ExercisesExercises

1. a line 2. a plane

3. three segments 4. three rays

5. a right angle 6. two acute angles

7. an obtuse angle 8. a pair of complementary angles

9. two pairs of supplementary angles

10. two lines 11. a plane

12. three segments 13. five rays

14. a right angle 15. two acute angles

16. two obtuse angles

17. a pair of complementary angles

18. two pairs of supplementary angles

Use the figure for Exercises 19–26. Write true or false. If a statement is false,rewrite it so it is true.

19. �QUR is an obtuse angle.

20. �4 and �2 are supplementary.

21. �1 and �6 are supplementary.

22. �3 and �1 are complementary.

23. If m�1 � 35°, then m�6 � 40°.

24. If m�SUN � 150°, then m�SUR � 150°.

25. If m�1 � x° , then m�PUQ � 180° � x° .

26. m�1 � m�3 � m�5 � m�6 � 180°.

27. Reasoning Two complementary angles have a ratio of 1:2. What is the measure of each angle?

KEYWORD: MT8CA Parent

KEYWORD: MT8CA 8-1

GUIDED PRACTICE

PRACTICE AND PROBLEM SOLVING

INDEPENDENT PRACTICE

Extra PracticeSee page EP16.

See Example 2

See Example 1

See Example 2

See Example 1

50°40°

60°30°

CaliforniaStandards Practice

Preparation for MG3.1

8-1 Points, Lines, Planes, and Angles 381

NS2.1, MG1.2, Prep for MG3.1

33. Multiple Choice When two angles are complementary, what is the sum of their measures?

90° 180° 270° 360°

34. Gridded Response �1 and �3 are supplementary angles. If m�1 � 63°, what is m�3?

Simplify each expression. Write your answer in exponential form. (Lesson 4-3)

35. m3 � m2 36. w � w6 37. 78 � 73 38. 116 � 119

39. Callie made a postcard that is 5 in. tall by 7 in. wide. A company would like tosell a poster based on the postcard. The poster will be 2 ft tall. How wide will theposter be? (Lesson 5-7)

Geography

Mapmakers often include a legend on the mapsthey create. The legend explains what each symbolor location on the map represents.

28. Name the geometric figure suggested by eachpart of the map.

a. City Hall and Gordon Middle School

b. Highway 80

c. the section of the road from the park to thepost office

d. the road from City Hall past the police station

29. A student rides her bike from Gordon MiddleSchool to City Hall. She then rides to the citypark, first passing through the intersection nearthe police station and then passing by theschool. List the segments on the map that represent her route.

30. Critical Thinking Name a line segment, a ray,and a line that include the same two locationson the map, but do not include the city park.

31. Write About It Explain why the roadfrom City Hall that goes past the police stationsuggests a ray named VX�� rather than a raynamed XV��.

32. Challenge What are all the possible names for the line suggested by IH-45?

City Hall

Post Office

Police Station

Gordon Middle School

City Park

MAP LEGEND

HWY 80

Construct Bisectors andCongruent Angles

Use with Lesson 8-1

Congruent figures have the same size and shape. Congruent angles have the same measure, and congruent segments have the same length.When you bisect a figure, such as a segment or an angle, you divide itinto two congruent parts. You can bisect segments and angles andconstruct congruent angles without using a protractor or ruler. Instead,you can use a compass and a straightedge.

Bisect a line segment.

a. Draw JS�� on a piece of paper.

b. Place the point of your compass on endpoint J and, using an opening that is greater than half the length of JS��, draw an arc that intersects JS��.

c. Using the same opening as you did in part b, place the point of your compass on endpoint S and draw an arc. This arc should intersect the first arc above and below JS��.

d. Draw a line to connect the intersections of the arcs. Label the intersection of JS�� and the line point K.

Use a ruler to measure JS��, J�K�, and K�S�. What do you notice?

This bisector of JS�� is a perpendicular bisector because all of theangles formed measure 90°.

Bisect an angle.

a. Draw an acute angle on a piece of paper. Label the vertex H.

b. Place the point of your compass on H and draw an arc through both sides of the angle. Label points Gand E where the arc crosses each side of the angle.

c. Without changing your compass opening, draw intersecting arcs from point G and point E. Label the point of intersection D.

d. Draw HD��.

Use your protractor to measure angles GHE, GHD, and DHE. What do you notice?

Activity

Step c Step b

Step d

GStep b

Step c

Step d

KEYWORD: MT8CA Lab8

CaliforniaStandards

MG3.1 Identify and constructbasic elements of geometric figures(e.g., altitudes, midpoints, diagonals,angle bisectors, and perpendicularbisectors; central angles, radii,diameters, and chords of circles) byusing a compass and straightedge.

Think and Discuss

Try This

Construct congruent angles.

a. Draw an angle on your paper. Label the vertex B.

b. To construct a congruent angle, begin by drawing a ray, and label its endpoint C.

c. With your compass point on B, draw an arc that intersects both sides of your angle. Label the points of intersection A and M.

d. With the same compass opening, place the compass point on C and draw an arc through the ray. Label point D where the arc crosses the ray.

e. Set your compass opening to AM.

f. With the same compass opening, place your compass point on D, and draw another arc intersecting the first arc. Label the intersection F. Draw CF��.

Use your protractor to measure �ABM and �FCD. What do you notice?

1. How many bisectors would you use to divide an angle into four equal parts?

2. An 88° angle is bisected, and then each of the two angles formed are bisected. What is the measure of each of the smaller angles formed?

Use a compass and a straightedge to perform each construction.

1. Bisect a line segment.

2. Trace and then bisect angle GOB.

3. Construct an angle congruent to angle GOB.

8-1 Hands-On Lab 383

CaliforniaStandards

MG3.6 Identify elements ofthree-dimensional geometric objects(e.g., diagonals of rectangular solids)and describe how two or moreobjects are related in space (e.g.,skew lines, the possible waysthree planes might intersect).

Who uses this? Some artists, such as SolLeWitt, base their designs on the differentways that lines and planes can intersect.

Vocabularyparallel linesperpendicular linesskew linesparallel planesperpendicular planes

The table describes some ways in which two lines may be related to each other.

arelines in the sameplane that neverintersect.

intersect to form 90°angles, or rightangles.

are linesthat lie in differentplanes. They areneither parallel norintersecting.

Line WX is parallel toline YZ.

WX�� YZ��

Line RS is perpendicularto line TU.

RS�� ⊥ TU��

Line AB and line ML areskew lines.

AB�� and ML�� are skew.

Skew lines

Perpendicular lines

Parallel lines

Identifying Lines in Space

Identify two lines that have the givenrelationship.

parallel lines

JM��⏐⏐KL�� The lines are in the same planeand do not intersect.

perpendicular lines

JK�� ⊥ KL�� The lines intersect to form 90° angles.

skew lines

JN�� and ML�� The lines lie in different planes. Theyare neither parallel nor intersecting.

E X A M P L E 1

Two lines may beperpendicular eventhough they do notappear to beperpendicular. Use themarkings on thedrawings and read theproblem carefully todetermine which linesare perpendicular.

The red arrows onlines WX and YZshow that the linesare parallel.

Identifying Planes in Space

Identify two planes that appear to have the given relationship.

perpendicular planes

plane ABE ⊥ plane EBC The planes intersect to form right angles.

parallel planes

plane ABC⏐⏐plane DEF The planes do not intersect.

neither perpendicular nor parallel

plane ABE and plane DAC The planes intersect, but they do notform right angles.

The intersections of planes may form parallel or perpendicular lines. In the figure, planes Aand B are parallel. Plane C intersects planes Aand B, forming a pair of parallel lines, � and �.

2E X A M P L E

8-2 Geometric Relationships 385

Think and Discuss

1. Give an example of parallel lines, perpendicular lines, and skewlines found in your classroom.

2. Determine whether two lines must be parallel if they do notintersect. Explain.

3. Explain whether two lines can be both parallel and skew.

4. Describe a real-world example of parallel planes.

areplanes that do notintersect.

are planes thatintersect to form rightangles.

Plane M is parallel toplane N.

Plane M⏐⏐plane N.

Plane R is perpendicularto plane S.

Plane R ⊥ plane S.

Perpendicular planes

Parallel planes

A plane may benamed by any threepoints in the planethat are not on thesame line.

Like lines, planes may be parallel or perpendicular, but they cannotbe skew.

Identify two lines that have the given relationship.

1. perpendicular lines

2. skew lines

3. parallel lines

4. parallel planes

5. perpendicular planes

6. neither parallel nor perpendicular

Identify two lines that have the given relationship.

7. parallel lines

8. skew lines

10. parallel planes

12. neither parallel nor perpendicular

Describe each pair of lines as parallel, perpendicular, skew, or none of these.

13. 14. 15.

16. Capitol Street intersects 1st, 2nd, and 3rd Avenues, which are parallel to each other. West Street and East Street are perpendicular to 2nd Avenue.

a. Draw a map showing the six streets.

b. Suppose East and West Streets were perpendicular to Capitol Street rather than 2nd Avenue. Draw a map showing the streets.

KEYWORD: MT8CA 8-2

GUIDED PRACTICE

See Example 2

See Example 1

See Example 2

See Example 1

8-2 Geometric Relationships 387

AF2.2, AF4.2, MG3.6

27. Multiple Choice Which of the following line segments appears to be skew to MR��?

LQ�� RQ��

QT�� SR��

28. Multiple Choice Which plane appears to be parallel to plane LPN?

plane MNS plane QLM plane TPN plane QTS

Simplify. (Lesson 4-4)

29. (6m3)2 30. (�2a3b5)4 31. (3x4y)3 32. (�2p7q2r 3)3

33. The table shows the number of Calories in melon slices of various weights. Determine whether the data set shows direct variation. (Lesson 7-9)

The lines in the figure intersect to form a rectangular box.Use the figure for Exercises 17–21.

17. Name all the lines that are parallel to AD��.

18. Name all the lines that are perpendicular to FG��.

19. Name two lines that are skew lines.

20. Name all the lines that are not parallel to and do notintersect DH��.

21. Name all the planes that are perpendicular to plane ABC.

22. Science Calcite is one of the most commonminerals on Earth. Name the pairs of planes thatappear to be parallel in the calcite crystal at right.

23. Reasoning Is it possible for three planes tointersect in a single straight line? If so, make asketch. If not, explain why such an intersection is not possible.

24. What’s the Error? A student identified a pair oflines in the classroom and claimed that they wereboth perpendicular and skew. Explain the error.

25. Write About It Explain the similarities and differences between parallellines and parallel planes.

26. Challenge Lines p, q, and r are in a plane. Lines p and q are parallel, and line r intersects line p. Does line r intersect line q? Explain.

Weight (oz) 3.5 7 10.5 14

Calories (Cal) 25 50 75 100

CaliforniaStandards

Review of Grade 6MG2.2 Use the properties

of complementary andsupplementary angles and the sum of the angles of a triangle tosolve problems involving anunknown angle.Also covered: 6MG2.1

You can use what you know about complementary andsupplementary angles to find missing angle measurements.

Finding Angle Measures

Use the diagram to find each anglemeasure.

If m�2 � 75�, find m�4.

m�1 � 180° � 75° �2 and �1 are supplementary.

� 105°

m�4 � 180° � 105° �1 and �4 are supplementary.

� 75°

So m�2 � m�4, or �2 � �4.

If m�3 � x�, find m�1.

m�4 � 180° � x° �3 and �4 are supplementary.

m�1 � 180° � m�4 �1 and �4 are supplementary.

� 180° � (180° � x°) Substitute 180° � x° for m�4.

� 180° � 180° � x° Distributive Property

� x° Simplify.

So m�1 � m�3, or �1 � �3.

The angles in Example 1 are examples of adjacent angles and verticalangles. These angles have special relationships because of their positions.

• have a common vertex and a common side, butno common interior points. Angles 1 and 2 in the diagram aboveare adjacent angles.

• are the nonadjacent angles formed by twointersecting lines. Angles 1 and 3 are vertical angles. In Example 1,you found that vertical angles have the same measure, so verticalangles are congruent.

Vertical angles

Adjacent angles

E X A M P L E 1

Congruent angleshave the samemeasure. The symbolfor congruence is �,which is read as “iscongruent to.”

Who uses this? Submarinebuilders use parallel mirrorswhen making periscopes. (See Exercise 24.)

Finding Angle Measures of Parallel Lines Cut by Transversals

In the figure, line a⏐⏐line b. Find the measureof each angle.

m�4 � 74� Corresponding angles are congruent.

m�3 � 74° � 180� �3 is supplementary to the 74� angle.

� 74� � 74� Subtract 74� from both sides.�� m�3 � 106� Simplify.

m�5 � 106� �3 and �5 are alternate interior angles, sothey are congruent.

If two parallel lines are intersected by a transversal, • corresponding angles are congruent, • alternate interior angles are congruent, • and alternate exterior angles are congruent.

If the transversal is perpendicular to the parallel lines, all of theangles formed are congruent 90� angles.

Alternate interiorangles

Alternate exteriorangles

Correspondingangles

Think and Discuss

1. Tell how many different angles would be formed by a transversalintersecting three parallel lines. How many different anglemeasures would there be?

2. Explain how a transversal could intersect two other lines so thatcorresponding angles are not congruent.

2E X A M P L E

8-3 Angle Relationships 389

A is a line that intersects two or more lines that lie in the sameplane. Transversals to parallel lines form angle pairs with special properties.

transversal

PROPERTIES OF TRANSVERSALS TO PARALLEL LINES

Use the diagram to find each angle measure.

1. If m�3 � 105°, find m�1.

2. If m�2 � x°, find m�4.

In the figure, line m⏐⏐line n. Find themeasure of each angle.

3. �1 4. �4

5. �6 6. �7

Use the diagram to find each angle measure.

7. If m�2 � 126°, find m�4.

8. If m�1 � b°, find m�3.

In the figure, line p⏐⏐line q. Find the measure of each angle.

9. �1

10. �4

11. �6

12. �7

In the figure, line t⏐⏐line s.

13. Name all angles congruent to �1.

14. Name all angles congruent to �2.

15. Name three pairs of supplementary angles.

16. Which line is the transversal?

17. If m�4 is 51�, what is m�2?

18. If m�7 is 116�, what is m�3?

19. If m�5 is x�, what is m�2?

20. Reasoning Two parallel lines are cut by a transversal. Can you determine the measures of all the angles formed if given only one angle measure? Explain.

KEYWORD: MT8CA 8-3

GUIDED PRACTICE

See Example 2

See Example 1

See Example 2

See Example 1

Review of 6MG2.1 and 6MG2.2

Draw a diagram to illustrate each of the following.

21. line p⏐⏐line q⏐⏐line r and line s is a transversal to lines p, q, and r

22. line m⏐⏐line n and transversal h with congruent angles �1 and �3

23. line h⏐⏐line j and transversal k with eight congruent angles

24. Science A periscope contains twoparallel mirrors that face each other.With a periscope, a person in asubmerged submarine can seeabove the surface of the water.

a. Name the transversal in the diagram.

b. If m�1 � 45°, find m�2, m�3, and m�4.

25. What’s the Error? Line a is parallel to line b. Line c is perpendicular to line b. Line c forms a 60° angle with line a. Why is this figure impossible to draw?

26. Write About It Choose an example of abstract art or architecture with parallellines. Explain how parallel lines, transversals, or perpendicular lines are used inthe composition.

27. Challenge In the figure, �1, �4, �6, and �7 are all congruent, and �2, �3, �5, and �8 are all congruent. Does this mean that line s is parallel to line t? Explain.

�1 � �2�3 � �4

8-3 Angle Relationships 391

Rev. of 6MG2.2, AF3.1, AF3.3

28. Multiple Choice Two parallel lines are intersected by a transversal. Themeasures of two corresponding angles that are formed are each 54°. What arethe measures of each of the angles supplementary to the corresponding angles?

36° 72° 108° 126°

29. Extended Response Suppose a transversal intersects two parallel lines. One angle that is formed is a right angle. What are the measures of theremaining angles? What is the relationship between the transversal and the parallel lines?

Graph each linear function. (Lesson 7-3)

30. y � �x � 1 31. y � 2x

Create a table for each quadratic function, and use it to graph the function. (Lesson 7-4)

32. y � �x 2 � 1 33. y � 2x 2 � x

CaliforniaStandards

MG3.3 Know andunderstand the Pythagoreantheorem and its converse and use itto find the length of the missingside of a right triangle andlengths of other line segmentsand, in some situations, empiricallyverify the Pythagorean theorem bydirect measurement.Also covered: Review of

6MG2.2

VocabularyTriangle Sum Theoremacute triangleright triangleobtuse triangleequilateral triangleisosceles trianglescalene trianglemidpointaltitude

If you tear off two corners of a triangle and place them next to thethird corner, the three angles seem to form a straight angle. You canalso show this in a drawing.

Draw a triangle and extend one side. Then draw a line parallel to the extended side, as shown below. The three angles in the triangle can bearranged to form a straight angle, or 180°.

Two sides of the triangle are transversals to the parallel lines. So both the blue angles and red anglesare pairs of alternate interior angles.

An has 3 acute angles. A has 1 right angle. An has 1 obtuse angle.

Finding Angles in Acute, Right, or Obtuse Triangles

Find x in the acute triangle.

63° � 42° � x ° � 180° Triangle Sum Theorem

105 � x � 180� 105 � 105 Subtract 105 from

�� x � 75 both sides.

Find y in the right triangle.

37° � 90° � y° � 180° Triangle Sum Theorem

127 � y � 180� 127 � 127 Subtract 127

�� y � 53 from both sides.

obtuse triangleright triangleacute triangle

TRIANGLE SUM THEOREMWords Numbers Algebra

79°43° s°t°

r°The interior anglemeasures of a triangleadd to 180°.

43° � 58° � 79° � 180° r° � s° � t ° � 180°

37°y°

E X A M P L E 1

8-4 Triangles

Who uses this? Triangles are often usedby designers to create unique flag designs.(See Exercise 22.)

An has 3 congruent sides and 3 congruent angles. Anhas at least 2 congruent sides and 2 congruent angles. A

has no congruent sides and no congruent angles.

Finding Angles in Equilateral, Isosceles, or Scalene Triangles

Find the angle measures in the equilateral triangle.

3m° � 180° Triangle Sum Theorem

3m� � �

1380� Divide both sides by 3.

m � 60

All three angles measure 60°.

Find the angle measures in the scalene triangle.

2p° � 3p° � 4p° � 180° Triangle Sum Theorem

9p � 180 Simplify.

�99p� � �

1980� Divide both sides by 9.

p � 20

The angle labeled 2p° measures 2(20°) � 40°, the angle labeled 3p° measures 3(20°) � 60°, and the angle labeled 4p° measures 4(20°) � 80°.

Finding Angles in a Triangle That Meets Given Conditions

The second angle in a triangle is twice as large as the first. Thethird angle is half as large as the second. Find the angle measuresand draw a possible figure.

Let x ° � first angle measure. Then 2x ° � second angle measure, and�12�(2x)° � x ° � third angle measure.

x° � 2x° � x° � 180° Triangle Sum Theorem

�44x� � �

1480�

Simplify, then divide bothsides by 4.

x � 45

Two angles measure 45° and one angle measures 90°. The triangle hastwo congruent angles. The triangle is an isosceles right triangle.

The of a segment is the point that dividesthe segment into two congruent segments. An

of a triangle is a perpendicular segmentfrom a vertex of the triangle to the line containingthe opposite side.

altitude

midpoint

scalene triangleisosceles triangle

equilateral triangle

2E X A M P L E

m° m°

3p° 2p°

8-4 Triangles 393

E X A M P L E 3

altitude

Reasoning

1. Find q in the acute triangle.

2. Find r in the right triangle.

3. Find s in the obtuse triangle.

KEYWORD: MT8CA 8-4

GUIDED PRACTICESee Example 1

Think and Discuss

1. Explain whether a right triangle can be equilateral. Can it beisosceles? scalene?

2. Explain whether a triangle can have 2 right angles. Can it have 2 obtuse angles?

31° r°

70° 33°

Finding the Length of a Line Segment

In the figure, M is the midpoint of PQ��

and RM�� is an altitude of �PQR. Find the length of RM��.

Step 1 Find the length of MQ��.

MQ � �12�PQ M is the midpoint of PQ��.

� �12�(24) � 12

Step 2 Use the Pythagorean Theorem and �RMQ. Let RM � a and MQ � b.

a2 � b2 � c 2 Pythagorean Theorem

a2 � 122 � 202 Substitute 12 for b and 20 for c.

a2 � 144 � 400 Simplify the powers.� 144 �144 Subtract 144 from each side.

�� a2 � 256a � 16 Find the square root.

The length of RM�� is 16 m, or RM is 16 m.

E X A M P L E 4

Review of 6MG2.2; MG3.3

MQ�� is the name of aline segment.MQ is the length ofthe segment.

9. Find r in the acute triangle.

10. Find s in the right triangle.

11. Find t in the obtuse triangle.

12. Find the angle measures in the equilateral triangle.

13. Find the angle measures in the isosceles triangle.

14. Find the angle measures in the scalene triangle.

15. The second angle in a triangle is five times as large as the first. The third angle is two-thirds as large as the first. Find the angle measures and draw a possible figure.

16. In the figure, K is the midpoint of JL�� and KM�� isperpendicular to JL��. Find the length of JM��.

Find the value of each variable. Classify the triangle by its sides and angles.

17. 18. 19.

See Example 2

See Example 3

See Example 2

See Example 3

See Example 1

c° c°

a°5d°

y°5y° w°

(w + 15)°45°

s°r°

23° 71°

m° m°

25° 40°

7g°2g°

8-4 Triangles 395

4. Find the angle measures in the equilateral triangle.

5. Find the angle measures in the isosceles triangle.

6. Find the angle measures in the scalene triangle.

7. The second angle in a triangle is half as large as the first. The third angle isthree times as large as the second. Find the angle measures and draw apossible figure.

8. In the figure, DB�� is a diagonal of rectangle ABCD.Find the length of A�B�.

50 ft14 ft

24 in.KJ

See Example 4

For more onclassifying triangles,see page SB16.

20. Reasoning Can an acute isosceles triangle have two angles that measure40°? Explain.

21. Triangle LMN is an obtuse triangle and m�L � 25°. �M is the obtuse angle.What is the largest m�N can be to the nearest whole degree?

22. Social Studies American Samoa is aterritory of the United States made up of agroup of islands in the Pacific Ocean, abouthalfway between Hawaii and New Zealand.The flag of American Samoa is shown.

a. Find the measure of each variable in the blue triangles.

b. Use your answers to part a to find m, x, and z.

c. Classify the triangles in the flag by their sides and angles.

23. Choose a Strategy Which of the following sets of angle measures can be used to create an isosceles triangle?

45°, 45°, 95° 49°, 51°, 80° 27°, 27°, 126° 35°, 55°, 100°

24. Write About It Explain how to cut a square or an equilateral triangle in half to form two identical triangles. What are the angle measures in the resulting triangles in each case?

25. Challenge Find x, y, and z in the figure shown.

Rev. of 6MG2.2; NS2.4, MG3.6

26. Multiple Choice Which type of triangle can be constructed with a 50° angle between two 8-inch sides?

Equilateral Isosceles Scalene Obtuse

27. Short Response Two angles of a triangle are 45° and 30°. What is the measure of the third angle? Is the triangle acute, right, or obtuse?

Each square root is between two integers. Name the integers. (Lesson 4-7)

28. �42� 29. �71� 30. �35� 31. �296�

Identify two lines that have the given relationship. (Lesson 8-2)

32. skew lines

33. parallel lines

z°x°y°

20°40°

110°110°

Follow these steps to construct the altitude from P to QR�� of �PQR.

Identify andConstruct Altitudes

Use with Lesson 8-4

Think and Discuss

Try This

Recall that an altitude of a triangle is a perpendicular segment from avertex of the triangle to the line containing the opposite side. Analtitude can be inside, outside, or on the triangle. You can use acompass and straightedge to construct an altitude of a triangle.

Activity

CaliforniaStandards

MG3.1 Identify and constructbasic elements of geometric figures(e.g., altitudes, midpoints, diagonals,angle bisectors, and perpendicularbisectors; central angles, radii,diameters, and chords of circles) byusing a compass and straightedge.

a. Use your straightedge to extend QR��.

c. Draw arcs from points X and Y using thesame compass opening. Label theintersection of the arcs point Z.

b. Place the point of your compass at P. Drawan arc that intersects QR�� at points X and Y.

d. Place your straightedge so that it passesthrough points P and Z. Draw a segmentfrom P to QR��. This is the required altitude.

1. How could you check that the altitude is perpendicular to QR��?

2. How many different altitudes can you construct for any given triangle?

Copy each triangle. Then construct the altitude from A to BC��.

1. 2. 3.

KEYWORD: MT8CA Lab8

CaliforniaStandards

MG3.2 Understand and usecoordinate graphs to plot simplefigures, determine lengths and areasrelated to them, and determine theirimage under translations andreflections.Also covered: AF3.3

8-5 CoordinateGeometry

In computer graphics, a coordinate system is used to create images, from simple geometric figures to realistic figures used in movies.

Properties of the coordinate plane can be used to find information about figures in the plane, such as whether lines in the plane are parallel.

Vocabularypolygonquadrilateraltrapezoidparallelogramrectanglerhombussquare

Finding Perpendicular and Parallel Lines

Which lines are parallel? Which lines are perpendicular? Explain.

Step 1 Find the slope of each line.

slope of PQ�� 32�

slope of RS�� 43�

slope of AB�� 32�

slope of PA�� 22� or �1

slope of GH�� 43�

slope of XY�� 87�

Step 2 Compare the slopes.

PQ��⏐⏐AB�� The slopes are equal: �32� � �

32�.

RS�� ⊥ GH�� The slopes have a product of �1: �43� � �

�43� � �1.

2–2–4

If a line has slope

�, then a line

perpendicular to it

has slope ��ba�.

Slopes of Parallel and Perpendicular Lines

• Any two nonvertical lines with equal slopes are parallel. Any twovertical lines are parallel.

• Any two nonvertical lines whose slopes have a product of �1 areperpendicular. Vertical and horizontal lines are perpendicular.

1E X A M P L E

Who uses this? Graphic designers usecoordinate geometry to create movie images.

8-5 Coordinate Geometry 399

A is a closed plane figure formed by three or more linesegments called sides. Each side meets exactly two other sides, oneon each end, in a common endpoint. are polygonswith four sides and four angles. Quadrilaterals with certain propertiesare given additional names.

Quadrilaterals

polygon

Quadrilaterals

Parallelograms2 pairs of parallel sides

Trapezoidsexactly 1 pair of

parallel sides

Rhombuses4 congruent sides

Rectangles4 right angles

Squares4 congruent sides

4 right angles

E X A M P L E 2 Using Coordinates to Classify Quadrilaterals

Graph the quadrilaterals with the given vertices. Give all of thenames that apply to each quadrilateral.

J(1, 2), K(4, 2), P(�1, 2), Q(2, 1),L(4, �1), M(1, �1) R(�1, �2), S(�3, 0)

2 pairs of parallel sides 1 pair of parallel sides

4 right angles trapezoid

parallelogram, rectangle, rhombus, square

Undefinedslope

Slope = 0

Slope = 0M

–1 2 6

Slope = – 13

Slope = 1

Slope = –1

For more on classifyingquadrilaterals, seepage SB16.

Reasoning

1. Which lines are parallel? Explain.

2. Which lines are perpendicular? Explain.

Graph the quadrilaterals with the given vertices.Give all of the names that apply to eachquadrilateral.

3. D(�3, �2), E(�3, 3), F(2, 3), G(2, �2)

4. R(�4, �1), S(�2, 2), T(4, 2), V(5, �1)

Find the coordinates of the missing vertex.

5. rhombus ABCD with A(2, 3), B(3, 1), and D(1, 1)

6. square JKLM with J(�3, 1), K(0, 1), and L(0, �2)

KEYWORD: MT8CA 8-5

See Example 2

Finding the Coordinates of a Missing Vertex

Find the coordinates of the missing vertex of square ABCD withA(4, 0), B(0, 4), and C(�4, 0).

Step 1 Graph and connect thegiven points.

Step 2 Complete the figure to find the missing vertex.A�B� has a slope of �1, so draw CD�� with a slope of�1. B�C� has a slope of 1, sodraw AD�� with a slope of 1.The rays intersect at (0, �4).

The coordinates of D are (0, �4).

Think and Discuss

1. Explain how you can use slopes to classify a quadrilateral.

3E X A M P L E

See Example 3

–2 2

AF3.3, MG3.2

8-5 Coordinate Geometry 401

7. Which lines are parallel? Explain.

8. Which lines are perpendicular? Explain.

Graph the quadrilaterals with the given vertices.Give all of the names that apply to eachquadrilateral.

9. D(�4, 3), E(4, 3), F(4, �5), G(�4, �5)

10. W(�2, 1), X(�2, �2), Y(4, 1), Z(0, 2)

11. rectangle ABCD with A(�3, 3), B(4, 3), and D(�3, �1)

12. trapezoid JKLM with J(�1, 5), K(2, 3), and L(2, 1)

Draw the line through the given points and find its slope.

13. A(1, 0), B(2, 3) 14. C(�3, 0), D(�3, �4)

15. G(4, �2), H(�1, �2) 16. E(�2, 1), F(3, �2)

17. A line passes through the coordinates P(1, 3) and Q(�2, �3). Identify the slope of PQ��. Then name two coordinates and the slope of a line perpendicular to PQ��.

18. On a coordinate grid draw a line m with slope 0 and a line n with slope �1.Then draw three lines through the intersection of lines m and n that haveslopes between 0 and �1.

19. On a coordinate grid draw a line s with slope 0 and a line t with slope 1. Then draw three lines through the intersection of lines s and t that have slopes between 0 and 1.

Tell if each statement is true or false. If it is false, explain.

20. The slope of a line through the origin is always defined.

21. Opposite sides of a rhombus have the same slope.

22. All of the adjacent sides of quadrilaterals have slopes with a product of �1.

23. All parallelograms have two pairs of lines with the same slope and adjacent sides that have slopes with a product of �1.

24. A trapezoid has two pairs of sides that have the same slope.

25. The slope of a horizontal line is always 0.

26. Reasoning Triangle LMN has vertices at L(�2, 2), M(0, 0), and N(�5, �1).What kind of triangle is it? Explain.

See Example 1

See Example 2

See Example 3

–2 4–4

27. AB��⏐⏐CD�� and the slope of AB�� is undefined. What can you tell about the slope of CD��? Explain.

28. Reasoning Square ABCD has vertices at (1, 2) and (1, �2). Find thepossible coordinates of the two missing vertices to create the square withthe least area. Justify your solution.

Identify and name each figure.

29. This figure has two sides with undefined slopes.

30. This figure has a side with a slope of �1.

31. This figure has a side with a slope of 3.

32. This figure has a side with a slope of �13�.

33. What’s the Question? Points P(3, 7),Q(5, 2), R(3, �3), and S(1, 2) form thevertices of a polygon. The answer is thatthe segments are not perpendicular. What is the question?

34. Write About It Explain how using different points on a line to find theslope affects the answer.

35. Challenge Use a square in a coordinate plane to explain why a line withslope 1 makes a 45° angle with the x-axis.

4 5321

�3�4�5 �2�1

�2�1

�3�4�5

NS1.3, AF3.3, AF4.2

36. Multiple Choice A right triangle has vertices at (0, 0), (0,4), and (10, 4). What is the slope of the hypotenuse?

2.5 2 1.8 0.4

37. Gridded Response Find the slope of the line that crosses through the points A(2, 4) and B(�1, 5).

Find each number. (Lesson 6-4)

38. 60% of what number is 12? 39. 112 is 80% of what number?

40. 30 is 2% of what number? 41. 90% of what number is 18?

Determine whether the data sets show direct variation. (Lesson 7-9)

Amount of Chicken per Serving

Amount (lb) 1 2 3 4 5

Servings 2 4 6 8 10

Pentagon ABCDE has been divided into three triangular regions by drawing all possible diagonals from one vertex.

1. Find each of the following:m�1 � m�2 � m�3 � ___?____

m�4 � m�5 � m�6 � ___?____

m�7 � m�8 � m�9 � ___?____

2. Add the three expressions.

m�1 � m�2 � m�3 � m�4 � m�5 � m�6 � m�7 � m�8 � m�9 � ___?____

3. Determine the sum of the interior angle measures of pentagon ABCDE(that is, m�EAB � m�B � m�BCD � m�CDE � m�E � ___?____).

4. Complete the table. Use sketches to illustrate your answers.

1. What information did you use to find the sums in Step 1?

2. Write a formula for the sum of the interior angle measures of an n-gon.

Find the sum of the measures of the angles of each polygon.

1. an octagon 2. a decagon (10 sides) 3. a nonagon (9 sides)

Angles in Polygons

Use with Lesson 8-5

Think and Discuss

Try This

KEYWORD: MT8CA Lab8

A diagonal of a polygon is a segment that connects any twononconsecutive vertices of the polygon. By drawing all possiblediagonals from one vertex, you can find the sum of the interior angle measures of a polygon.

Activity

CaliforniaStandards

MG3.1 Identify and constructbasic elements of geometricfigures (e.g., altitudes, midpoints,diagonals, angle bisectors, andperpendicular bisectors; central angles,radii, diameters, and chords of circles)by using a compass and straightedge.

Number of Number of Sum of anglePolygon sides triangular regions measures

triangle 1 180°

quadrilateral

pentagon 3 540°

hexagon

n-gon n

Quiz for Lessons 8-1 Through 8-5

8-1 Points, Lines, Planes, and Angles

1. two pairs of complementary angles

2. three pairs of supplementary angles

3. two right angles

Identify two figures that have the given relationship.

4. skew lines

5. parallel planes

In the figure, line m⏐⏐line n. Find the measure of each angle.

7. �1 8. �2 9. �3

8-4 Triangles

Find x in each triangle.

10. 11.

12. PR�� is a diagonal of rectangle PQRS. PQ is 28 ft and PR is 35 ft. Find the lengthof QR��.

8-5 Coordinate Geometry

Graph the quadrilaterals with the given vertices. Give all of the names that apply toeach quadrilateral.

13. A(�2, 1), B(3, 2), C(2, 0), D(�3, �1) 14. P(�3, 4), Q(2, 4), R(2, �1), S(�3, �1)

15. square ABCD with A(�1, 1), B(2, 1), and C(2, �2)

16. parallelogram PQRS with P(3, 3), Q(4, 2), and R(2, �2)

75° 55°

x°60°

x° x°

In the figure, �1 and �2 are complementary, and �1 and �5 are supplementary. If m�1 � 60°, find m�3 � m�4.

In triangle ABC, m�A � 35° and m�B � 55°. Is triangle ABC a right triangle?

The second angle in a quadrilateral is eighttimes as large as the first angle. The third angleis half as large as the second. The fourth angleis as large as the first angle and the secondangle combined. Find the approximate anglemeasures in the quadrilateral.

Parallel lines m and n are intersected by atransversal, line p. The acute angles formed by line m and line p measure 45°.Find the measure of the obtuse anglesformed by the intersection of line n and line p.

Write each problem in your own words. Check to make sure youhave included all of the information needed to solve the problem.

Understand the Problem• Restate the problem in your

own words

If you write a problem in your own words, you may understand itbetter. Before writing a problem in your own words, you may needto read it over several times—perhaps aloud, so you can hearyourself say the words.

Once you have written the problem in your own words, you maywant to make sure you included all of the necessary information tosolve the problem.

Focus on Problem Solving 405

55°35°A

CaliforniaStandards

MR1.1 Analyze problems byidentifying relationships, distinguishingrelevant from irrelevant information,identifying missing information,sequencing and prioritizinginformation, and observing patterns.Also covered: MG3.0

CaliforniaStandards

MG3.4 Demonstrate anunderstanding of conditions thatindicate two geometrical figuresare congruent and whatcongruence means about therelationship between the sidesand angles of the two figures.

Vocabularycorrespondencecongruent

E X A M P L E 1

Marks on the sides ofa figure can be usedto show congruence.K�M� � R�S� (1 mark)K�L� � R�Q� (2 marks)M�L� � S�Q� (3 marks)

Congruent Polygons

Who uses this?Biologists use congruenceto compare DNA profiles.

Below are the DNA profiles oftwo pairs of twins. Twins A andB are identical twins. Twins Cand D are fraternal twins.

A is a way of matching uptwo sets of objects. The bands of DNA in eachpair match up, or correspond. In the DNA ofthe identical twins, the corresponding bandsare the same.

figures have the same size and shape. If two polygons arecongruent, all of their corresponding sides and angles are congruent.

To write a congruence statement, the vertices in the second polygon have tobe written in order of correspondence with the first polygon.

Writing Congruence Statements

Write a congruence statement for the pair of congruent polygons.

�K corresponds to �R. �K � �R�L corresponds to �Q. �L � �Q�M corresponds to �S. �M � �S

The congruence statement is triangle KLM � triangle RQS.

Congruent

correspondence

CONGRUENT TRIANGLES

Corresponding CorrespondingDiagram Statement Angles Sides

�A � �D A�B� � D�E�

�ABC � �DEF �B � �E B�C� � E�F�

�C � �F A�C� � D�F�

94°21°

Write a congruence statement for the pair of congruent polygons.

�A corresponds to �H. �A � �H�B corresponds to �I. �B � �I�C corresponds to �J. �C � � J�D corresponds to �F. �D � �F�E corresponds to �G. �E � �G

The congruence statement is pentagon ABCDE � pentagon HIJFG.

Using Congruence Relationships to Find Unknown Values

In the figure, quadrilateral PQSR � quadrilateral WTUV.

Find x. Find y.x � 5 � 12 P�R� � W�V� 6y � 24 W�T� � P�Q�

� 5 � � 5�66y� � �

264�

x � 7y � 4

Find z.132 � 11z �R � �V

�11312

� � �1111z

� Divide both sides by 11.

12 � z

Think and Discuss

1. Explain what it means for two polygons to be congruent.

2. Tell how to write a congruence statement for two polygons.

DE130° 100°

80°135°G

100°80°

2E X A M P L E

x � 518

(11z)°12

T6y75°

Divide bothsides by 6.

Subtract 5 fromboth sides.

8-6 Congruent Polygons 407

The vertices in apolygon are writtenin order around thepolygon starting atany vertex.

Write a congruence statement for each pair of congruent polygons.

In the figure, triangle ABC � triangle LMN.

3. Find q. 4. Find r. 5. Find s.

Write a congruence statement for each pair of congruent polygons.

In the figure, quadrilateral ABCD � quadrilateral LMNO.

8. Find m. 9. Find n. 10. Find p.

Find the value of each variable.

11. pentagon ABCDE � 12. hexagon ABCDEF �pentagon PQRST hexagon LMNOPQ

KEYWORD: MT8CA 8-6

GUIDED PRACTICE

See Example 2

See Example 1

A 58°

r � 9

N M25°

D 32°

123° G

92°144°

75°49°

92° 144°

75°49°

(180 � pA F

(180 � p)°

(n � 86)°

(11m)°

75°Y X

131° 11

L M(m � 75)°

n + 421

(80 � y)°

6x°6.5

T114°

z – 4.5

Find the value of each variable.

13. quadrilateral ABCD � 14. heptagon ABCDEFG �quadrilateral EFGH heptagon JKLMNOP

15. Right triangle PQR � right triangle STU. m�P � 28° and m�U � 90°. Find m�Q.

16. Write a Problem Write and solve a problem about the right triangles shown.

17. Write About It How can knowing that two polygons are congruent help you find angle measures of the polygons?

18. Challenge Triangle ABC � triangle LMNand A�E�⏐⏐B�D�. Find m�ACD.

(w � 88)°

102°123°

s � 55

t � 30

56 139°

P °x4

8-6 Congruent Polygons 409

AF3.3, MG3.2, MG3.4

19. Multiple Choice Triangle EFG �triangle JIH. Find the value of x.

x � 5.67 x � 30 x � 63 x � 71

20. Multiple Choice Triangle ABC � triangle JKL. m�A � 30° and m�B � 50°. Find m�K.

30° 50° 80° 100°

21. Gridded Response Quadrilateral ABCD � quadrilateral WXYZ. The length of A�B� � 21 and the length of W�X� � 7m. Find m.

22. A function has an output value of 1 when the input value is 2; it has anouput value of 2 when the input value is 4; and it has output value of �1when the input value is 2. Is this a linear function? (Lesson 7-3)

Find the coordinates of the missing vertex. (Lesson 8-5)

23. square PQRS with Q(�2, 1), R(�2, �1), and S(�4, �1)

24. rectangle LMNO with L(1, 3), M(3, 3), and N(3, �2)

9w 7.6z

CaliforniaStandards

MG3.2 Understand and usecoordinate graphs to plot simplefigures, determine lengths and areasrelated to them, and determinetheir image under translationsand reflections.

Vocabularytransformationimagetranslationreflectionrotation

E X A M P L E 1

The point that afigure rotatesaround may be onthe figure or awayfrom the figure.

Who uses this? Professional iceskaters perform transformations duringtheir routines.

Translation Reflection Rotation

A slides a A flips a A turns afigure along a line figure across a line to figure around a point, without turning it. create a mirror image. called the center of

rotation.

rotationreflectiontranslation

Transformations

In the photograph, Michelle Kwan is performing a layback spin. She is holding her body in one position while she rotates. This is an example of a transformation.

In mathematics, a changes the position ororientation of a figure. The resulting figure is the of theoriginal. Images resulting from the transformations described beloware congruent to the original figures.

Identifying Types of Transformations

Identify each type of transformation.

The figure slides along The figure flips acrossa straight line. the x-axis.It is a translation. It is a reflection.

imagetransformation

Graphing Translations on a Coordinate Plane

Graph the translation of �ABC 6 units right and 4 units down.

Each vertex is moved 6 units right and 4 units down.

Graphing Reflections on a Coordinate Plane

Graph the reflection of each figure across the indicated axis. Write the coordinates of the vertices of the image.

x-axis

The x-coordinates of thecorresponding vertices are thesame, and the y-coordinates ofthe corresponding vertices areopposites.

The coordinates of the vertices of triangle F�G�H� are F�(�3, �3),G�(1, �4), and H�(3, �1).

y-axis

The y-coordinates of thecorresponding vertices are thesame, and the x-coordinates ofthe corresponding vertices areopposites.

The coordinates of the vertices of figure A�B�C�D�E�F�G� are A�(5, �4), B�(3, �2), C�(3, �3), D�(1, �3), E�(1, �5), F�(3, �5), and G�(3, �6).

�2�4

B′ C′

6 units right

4 unitsdown

E X A M P L E 3

E X A M P L E

A’ is read “A prime”and is used torepresent the pointon the image thatcorresponds to point A of theoriginal figure.

F′G′

C′D′

E′F′

2 4�2

8-7 Transformations 411

Graphing Rotations on a Coordinate Plane

Triangle JKL has vertices J(�3, 1), K(�3, �2), and L(1, �2). Rotate�JKL 90° counterclockwise about the vertex J.

Notice that the corresponding sides,J�K� and J�K�’, make a 90° angle.

Also notice that vertex K is 3 unitsbelow vertex J, and vertex K’ is 3 units to the right of vertex J.

Think and Discuss

1. Describe a classroom situation that illustrates a translation.

2. Tell what happens to the x-coordinate and the y-coordinate after apoint is reflected across the x-axis.

E X A M P L E 4

rotate 90°

Graph each translation.

3. 2 units left and 3 units up 4. 3 units right and 4 units down

See Example 2

KEYWORD: MS7 Parent

KEYWORD: MS7 8-10

KEYWORD: MT8CA 8-7

�2 2 O 42

Graph the reflection of each figure across the indicated axis. Write thecoordinates of the vertices of the image.

5. x-axis 6. y-axis

7. Triangle LMN has vertices L(0, 1), M(�3, 0), and N(�2, 4). Rotate �LMN180° about the vertex L.

Graph each translation.

10. 5 units right and 1 unit down 11. 4 units left and 3 units up

Graph the reflection of each figure across the indicated axis. Write thecoordinates of the vertices of the image.

12. y-axis 13. x-axis

14. Triangle MNL has vertices M(0, 4), N(3, 3), and L(0, 0). Rotate �MNL 90°counterclockwise about the vertex L.

42–2

�4 O

See Example 3

See Example 4

INDEPENDENT PRACTICESee Example 1

See Example 2

See Example 3

See Example 4

8-7 Transformations 413

AF3.3, MG3.2, MG3.4

28. Multiple Choice In the figure shown, what will be the coordinates of point X after a translation 2 units down and 3 units right?

(0, 1) (1, 0) (�1, 0) (0, �1)

29. Short Response Triangle ABC has vertices A(�3, 1), B(0, 1), and C(0, 6). Rotate �ABC 90° clockwise around vertex B. Draw �ABC and its image.

Determine whether the rates of change are constant or variable. (Lesson 7-6)

30. 31.

Determine the missing measure in each set of congruent polygons. (Lesson 8-6)

32. 33.

�4 O

4 m5 m

3 mA C

4 m5 m

8 in.b

x �1 0 2 5 9

y 1 3 5 7 9

Draw the image of the parallelogram ABCD with vertices (�3, 0), (�4, 3), (1, 4),and (2, 1) after each transformation.

15. translation 1 unit up 16. reflection across the x-axis

Draw the image of the quadrilateral ABCD with vertices (1, 1), (2, 4), (4, 5), and (5, 3) after each transformation.

17. translation 5 units down 18. reflection across the y-axis

Give the coordinates of each point after a reflection across the given axis.

19. (�3, 2); x-axis 20. (1, 4); x-axis 21. (m, n); y-axis

22. (�2, 4); y-axis 23. (5, �2); y-axis 24. (m, n); x-axis

25. Write a Problem Write a problem involving transformations on acoordinate grid that result in a pattern.

26. Write About It Explain how each type oftransformation performed on the arrow would affect the direction the arrow is pointing.

27. Challenge A triangle has vertices (2, 5), (3, 7), and (7, 5). After a reflectionand a translation, the coordinates of the image are (7, �2), (8, �4), and(12, �2). Describe the transformations.

PRACTICE AND PROBLEM SOLVINGExtra Practice

See page EP17.

x �2 0 2 4 6

y 1 0 �1 �2 �3

2-7 Variables and Algebraic Expressions 415

Follow the steps below to transform a figure.

a. Place a rhombus on a coordinate plane. Trace the rhombus, and label the vertices.

b. Rotate the figure 90° clockwise about the origin.

c. Reflect the resulting figure across the x-axis. Draw the image and label the vertices.

d. Now place a rhombus in the same position as the original figure. Reflect the figure across the line y � x.

1. What do you notice about the images that result from the two transformations in parts b and c above and the image that results from the single transformation in part d above?

1. Place a pattern block on a coordinate plane. Trace the block and label the vertices. Perform two different transformations on the figure. Draw the image and label the vertices. Explain what single transformation of the original figure would result in the same image.

Describe two different ways to transform each figure from position A to position B.

Activity

Think and Discuss

Try This

You can use a coordinate plane when transforming a geometric figure.

CombineTransformations

KEYWORD: MT8CA Lab8

Use with Lesson 8-7

CaliforniaStandards

MG3.2 Understand and use coordinate graphs to plotsimple figures, determine lengths and areas related to them, anddetermine their image under translations and reflections.

A repeating pattern of plane figures thatcompletely covers a plane with no gapsor overlaps is a .

In a , a regularpolygon is repeated to fill a plane. Sincethe angle measures at each vertex of aregular tessellation must add to 360°,only three regular tessellations exist.

Equilateral triangles Squares Regular hexagons

6 � 60° � 360° 4 � 90° � 360° 3 � 120° � 360°

It is also possible to tessellate with polygons that are not regular. Sincethe angle measures of a triangle add to 180°, six triangles meeting ateach vertex will tessellate. The angle measures of a quadrilateral add to360°, so four quadrilaterals meeting at a vertex will tessellate.

Creating a Tessellation

Create a tessellation withquadrilateral ABCD.

There must be a copy of each angleof quadrilateral ABCD at every vertex.

regular tessellation

tessellation

E X A M P L E 1B

8-8 Tessellations

Alcazar Palace in Seville, Spain

CaliforniaStandards

Extension of MG3.2 Understandand use coordinate graphs to plotsimple figures, determine lengthsand areas related to them, anddetermine their image undertranslations and reflections.

Vocabularytessellationregular tessellation

Why learn this? Tessellations are often used in art and architecture. (See Exercises 14–16.)

Think and Discuss

1. Explain why a regular pentagon cannot be used to create a regulartessellation.

2. Describe the transformations used to make the tessellation fromthe modified figure in Example 2.

3. Explain why modifying the sides of a tessellating polygon asshown in Example 2 will always yield another tessellating figure.

2E X A M P L E

1. Create a tessellation with quadrilateral QRST.

2. Use rotations to create a variation of the tessellation in Exercise 1.

KEYWORD: MT8CA 8-8

GUIDED PRACTICE

See Example 2

See Example 1

Creating a Tessellation by Transforming a Polygon

Use rotations to create a variation of the tessellation in Example 1.Step 1: Find the midpoint of a side.

Step 2: Make a new edge for half of the side.

Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.

Step 4: Repeat with the other sides.

Step 5: Use the figure to make a tessellation.

8-8 Tessellations 417

8-8 ExercisesExercisesCaliforniaStandards Practice

Extension of MG3.2

Use each shape to create a tessellation.

5. 6. 7.

8. 9. 10.

11. A piece is removed from one side of a rectangle and translated to the opposite side. Will this shape tessellate?

12. A piece is removed from one side of a trapezoid and translated to the opposite side. Will this shape tessellate?

13. In a semiregular tessellation, two or more regular polygons are repeated to fill the plane and the vertices are all identical. Use each arrangement of regular polygons to create a semiregular tessellation.

a. b. c.

PRACTICE AND PROBLEM SOLVINGExtra Practice

See page EP17.

3. Create a tessellation with triangle PQR.

4. Use rotations to create a variation of the tessellation in Exercise 3.

See Example 2

See Example 1 Q

her ©

M. C. Escher created works of art by repeating interlocking shapes.He used both regular and nonregular tessellations. He often usedwhat he called metamorphoses, in which shapes change into othershapes. Escher used his reptile pattern in many hexagonaltessellations. One of the most famous is entitled simply Reptiles.

14. The steps below show the method Escher used to make abird out of a triangle. Use the bird to create a tessellation.

15. Reasoning What regular polygon do you think Escher used to begin Reptiles?

16. Challenge Create an Escher-like tessellation of your own design.

r ©20

Step 1 Step 2

Step 3 Step 4

8-8 Tessellations 419

KEYWORD: MT8CA Escher

17. Multiple Choice Which of the following shapes will NOT form a regular tessellation?

18. Short Response Which set of polygons will create a tessellation? Explain.

Write each number in scientific notation. (Lesson 4-5)

19. 3,400,000,000 20. 0.00000045 21. 28,000

Create a table for each cubic function, and use it to graph the function. (Lesson 7-5)

22. y � �13�x 3 23. y � �x 3 � 1 24. y � x 3 � 5

NS1.1, AF3.1, Ext. of MG3.2

Quiz for Lessons 8-6 Through 8-8

8-6 Congruent Polygons

In the figure, triangle ABC � triangle LMN.

1. Find q. 2. Find r. 3. Find s.

8-7 Transformations

Graph each transformation. Give the coordinates of the image’s vertices.

4. Translate triangle 5. Reflect the figure 6. Rotate triangle JKL 90°RST 5 units down. across the x-axis. clockwise about vertex L.

8-8 Tessellations

Copy the given figure and use it to create a tessellation.

135q°

r � 9

�2�4

Cloth Creations The Asante people of Ghana are known forweaving Kente cloth, a colorful textile based on repeatinggeometric patterns. Susan is using a coordinate plane to designher own Kente cloth pattern.

1. Susan starts with triangle ABCas shown. Explain how she canuse slopes to make sure thetriangle is a right triangle.

2. To begin the pattern, Susan uses transformations to make a row of triangles that are allcongruent to triangle ABC.Describe the transformations she should use to make these triangles.

3. Next, she extends the pattern by making additionalrows of triangles. The firsttriangle in Row 2 is shown.Complete the table by writingthe coordinates of the topvertex of each triangle in the pattern.

4. What patterns do you notice in the table?

5. Susan’s Kente cloth pattern is a tessellation of what types of figures?

–2–4–6–8–10

2 4 6O

–2–4–6–8–10

2 4 6O

–2–4–6–8–10

2 4 6O

Concept Connection 421

Row Top Vertex of Triangles in Row

Row 1 (�9, 2) (�4, 2) (1, 2) (6, 2)

Row 2 (�8, 4)

The object of this game is to create geometric figures. Each card in the deck shows a property of a geometric figure. To create a figure, you must draw a polygon that matches at least three cards in your hand. For example, if you have the cards “quadrilateral,” “a pair of parallel sides,” and “a right angle,” you could draw a rectangle.

A complete set of rules and playing cards is available online.

Coloring TessellationsTwo of the three regular tessellations—triangles and squares—can becolored with two colors so that no two polygons that share an edgeare the same color. The third—hexagons—requires three colors.

1. Determine if each semiregular tessellation can be colored withtwo colors. If not, tell the minimum number of colors needed.

2. Try to write a rule about which tessellations can be coloredwith two colors.

Polygon RummyPolygon Rummy

KEYWORD: MT8CA Games

Project CDGeometry

PROJECT

Materials • 3 sheets of white

paper• CD or CD-ROM• scissors• tape• markers• empty CD case

It’s in the Bag! 423

Make your own CD to record important factsabout plane geometry.

Fold a sheet of paper in half. Place a CD on topof the paper so that it touches the folded edge.Trace around the CD. Figure A

Cut out the CD shape, being careful to leave thefolded edge attached. This will create two paperCDs that are joined together. Cut a hole in thecenter of each paper CD. Figure B

Repeat steps 1 and 2 with the other two sheetsof paper.

Tape the ends of the paper CDs together tomake a string of six CDs. Figure C

Accordion fold the CDs to make a booklet.Write the number and name of the chapter onthe top CD. Store the CD booklet in an emptyCD case.

Taking Note of the MathUse the blank pages in the CD booklet to take notes on the chapter. Be sure to include definitions and sample problems that will help you review essential concepts about plane geometry.

Complete the sentences below with vocabulary words from the list above.

1. Lines in the same plane that never meet are called ___?___. Lines that intersect at 90° angles are called ___?___.

2. A quadrilateral with 4 congruent angles is a(n) ___?___ or a(n) ___?___. A quadrilateral with 4 congruent sides is a(n) ___?___ or a(n) ___?___.

Vocabulary

Points, Lines, Planes, and Angles (pp. 378–381)8-1

E X A M P L E EXERCISES

■ a linePossible answer: AD��

■ a planePossible answer: plane ADE

3. a ray

4. an acute angle

5. a pair of supplementary angles

. . . . . . . . . . . . 379

. . . . . . . . . . 392

. . . . . . . . 388

. . . . . . . . . . . . . . . 393

. . . . . . . . . . . . . . . . . . 379

. . 379

. . . . . . . . . . . . . 406

. . . . . . . . 406

. . . . . 393

. . . . . . . . . . . . . . . . . 410

. . . . . . . 393

. . . . . . . . . . . . . . . . . . . 378

. . . . . . . . . . . . . . 393

. . . . . . . . . . . 379

. . . . . . . . . 392

. . . . . . . . . . . 384

. . . . . . . . . . 399

. . . . . . . . . 385

. . . . 384

. . 385

. . . . . . . . . . . . . . . . . . 378

. . . . . . . . . . . . . . . 399

. . . . . . . . . . . 399

. . . . . . . . . . . . . . . . . . . . 378

. . . . . . . . . . . . . . 399

. . . . . . . . . . . . . . 410

. . . . . 416

. . . . . . . . . . . . . . 399

. . . . . . . . . . . . . 379

. . . . . . . . . . 392

. . . . . . . . . . . . . . . 410

. . . . . . . . 393

. . . . . . . . . . . . . . . 378

. . . . . . . . . . . . . 384

. . . . . . . . . . . . . . . . 399

. . . . . . . . . . 379

. . 379

. . . . . . . . . . . . 416

. . . . . . . . 410

. . . . . . . . . . . . 410

. . . . . . . . . . . . 389

. . . . . . . . . . . . . . 399

. . . . . . . . . 388vertical angles

Triangle Sum Theorem

trapezoid

transversal

translation

transformation

tessellation

supplementary angles

straight angle

square

skew lines

segment

scalene triangle

rotation

right triangle

right angle

rhombus

regular tessellation

reflection

rectangle

quadrilateral

polygon

perpendicular planes

perpendicular lines

parallel planes

parallelogram

parallel lines

obtuse triangle

obtuse angle

midpoint

isosceles triangle

equilateral triangle

correspondence

congruent

complementary angles

altitude

adjacent angles

acute triangle

acute angle

Prep for MG3.1

Angle Relationships (pp. 388–391)8-3

Line j⏐⏐line k. Find each angle measure.

■ m�1 m�1 � 143°

■ m�2 m�2 � 143° � 180°

� 143° � 143°�� m�2 � 37°

Line p⏐⏐line q. Find each angle measure.

9. m�1

10. m�2

11. m�3

12. m�4

13. m�5

143° 21

4 5114°

Triangles (pp. 392–396)8-4

■ Find n.

n° � 50° � 90° � 180°n � 140 � 180

� 140 � 140��

n � 40

14. Find m.

15. In the figure, D is the midpoint of AC�� and DB�� is perpendicular to AC��.Find the length of AB��.

Geometric Relationships (pp. 384–387)8-2

■ Identify two planes that appear to be parallel.

plane GCF ⏐⏐plane HDE

Identify two lines or planes that appearto have the given relationship.

6. skew lines

8. parallel planes

Study Guide: Review 425

The planes do not intersect.

10 cm12 cm

Rev. of 6MG2.2

Congruent Polygons (pp. 406–409)8-6

■ Triangle ABC � triangle FDE. Find x.

x � 4 � 4� 4 � 4

�� x � 8

Triangle JQZ � triangle VTZ.

19. Find x.

20. Find t.

21. Find q.

Transformations (pp. 410–414)8-7

■ Graph the translation.

Translate �ABC 1 unit right and 3 units down.

Draw the image of a triangle ABC withvertices (1, 1), (1, 4), and (3, 1) after eachtransformation.22. reflection across the x-axis

23. translation 5 units left

24. 180° rotation around (0, 0)

Tessellations (pp. 416–419)8-8

■ Create a tessellation with the figure.

Create a tessellation with each figure.

25. 26.

x � 4A

48°7.2

(3x)°

J 75°2q � 1 13

Coordinate Geometry (pp. 398–402)8-5

■ Graph the quadrilateral with the givenvertices. Give all the names that apply.D(�2, 1), E(2, 3), F(3, 1), G(�1, �1)

D�E�⏐⏐F�G�E�F� ⏐⏐G�D�D�E� ⊥ E�F�

Graph the quadrilaterals with the givenvertices. Give all the names that apply.

16. Q(2, 0), R(�1, 1), S(3, 3), T(8, 3)

17. K(2, 3), L(3, 0), M(2, �3), N(1, 0)

18. W(2, 2), X(2, �2), Y(�1, �3), Z(�1, 1)

parallelogram,rectangle

MG3.2, AF3.3

Ext. of MG3.2

Chapter 8 Test 427

In the figure, line m⏐⏐line n.

1. Name two pairs of supplementary angles.

2. Find m�1. 3. Find m�2.

4. Find m�3. 5. Find m�4.

Identify two lines or planes that appear tohave the given relationship.

6. skew lines

7. parallel planes

9. Two angles in a triangle have measures of 44° and 57°.What is the measure of the third angle?

10. What are the measures of the congruent angles in an isosceles triangle if the measure of the third angle is 102°?

11. In the figure, E is the midpoint of FG�� and DE�� is perpendicular to FG��. Find the length of DE��.

Give all of the names that apply to each figure.

12. 13.

Graph the quadrilateral with the given vertices. Give all of the names thatapply to each quadrilateral.

14. A(3,4), B(8,4), C(5,0), D(0,0) 15. K(�4,0), L(�2,5), M(2,5), N(4,0)

In the figure, quadrilateral ABCD � quadrilateral LMNO.

16. Find m.

17. Find n.

18. Find p.

Pentagon ABCDE has vertices A(1, �2), B(3, �1), C(7, �2), D(6, �4), and E(2, �5).Find the coordinates of the image of each point after each transformation.

19. rotation 90° around the origin, point E 20. reflection across the x-axis, point C

21. translation 6 units up, point B 22. reflection across the y-axis, point A

DAB || CDAD || BC

3 cm 3 cm

135°1 2

n � 6

(m � 45)°

KEYWORD: MT8CA Practice

1. A bird flies from the ground to the topof a tree, sits there and sings for a while,flies down to the top of a picnic table toeat crumbs, and then flies back to thetop of the tree to sing some more.Which graph best represents thissituation?

2. There are 36 poodles entered in a dogshow with a total of 144 entries. Whatpercent of the dogs are poodles?

10% 75%

25% 400%

3. Which of the following is NOT arational number?

��196� ��10�

�5.83� ��23�

4. Which statement describes theinequality x � �1?

A number is more than �1.

A number is at least �1.

A number is less than �1.

A number is at most �1.

5. An animal shelter needs to find homesfor 40 dogs and 60 cats. If 15% of thedogs are female and 25% of the catsare female, what percent of theanimals are female?

21% 40%

22% 42%

6. The line graph shows the activity of a savings account. What does the slope represent?

The number of months it takes tosave $4500

The initial amount in the account

The amount saved every 7 months

The amount saved every 2 months

7. Which equation represents a directvariation between x and y?

y � x � 2 y � 2 � x

y � �2x� y � 2x

8. Find the value of ⏐a⏐ � b2 whena � �3 and b � �5.

�28 �7

�22 4DB

Cumulative Assessment, Chapters 1–8Multiple Choice

4000300020001000

Months in plan

Savings Account Activity

20 4 61 3 5 7

9. George earns a weekly salary of $575plus a 6% commission on sales. Howmuch would he make in a week if hesold $7500 worth of merchandise?

$450 $2300

$1025 $2750

Gridded Response

10. What is the slope of the line graphed?

11. What is the output value for thefunction y � x3 � 2x when the inputvalue is 4?

12. What is the value of x when the slopeof the line connecting the points (�1, 4) and (x, 1) is ��

34�?

13. Bob is buying a $35 sweater that isdiscounted 20%. What is the total priceof the sweater when 5% sales tax isadded?

14. Marilyn wants to save $240. She hassaved $60 in the past 4 weeks. If shecontinues to save at the same rate,how many more weeks will it take herto save $240?

Short Response

15. Identify two lines that have the givenrelationship.

a. parallel lines

b. perpendicular lines

c. skew lines

16. Triangle ABC, with vertices A(2, 3), B(4, �5), and C(6, 8), is reflected acrossthe x-axis to form triangle A´B´C´.

a. On a coordinate grid, draw andlabel triangle ABC and triangleA´B´C´.

b. Give the coordinates of the verticesfor new triangle A´B´C´.

Extended Response

17. Paul Revere had to travel 3.5 miles toCharlestown from Boston by boat.Assume that from Charlestown toLexington he was able to ride a horsethat traveled at a rate of �

18� mile per

minute. His total distance traveled, y, isthe sum of the distance to Charlestownand the distance from Charlestown toLexington.

a. Write a linear equation that couldbe used to find the distance, y, PaulRevere traveled in x minutes.

b. What does the slope of the linerepresent?

c. Find the output value when theinput value is 0 and tell what thispoint represents in Paul Revere’sride.

d. Graph your equation from part a ona coordinate plane.

�4 �2

Cumulative Assessment, Chapters 1–8 429

Use logic to eliminate answer choicesthat are incorrect. This will help youmake an educated guess if you arehaving trouble with the question.

foundations of geometry - math with mr....

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