lesson ready to go on? skills intervention building …€¦ · · 2011-09-11ready to go on?...

Ready to Go On? Skills InterventionBuilding Blocks of Geometry8-1

LESSON

A point is an exact location. A line is a straight path that extendswithout end in opposite directions. A plane is a flat surface thatextends without end in all directions. A line segment is made oftwo endpoints and all the points between the endpoints. A ray hasone endpoint. From the endpoint, the ray extends without end inone direction only.

Identifying Points, Lines, and PlanesUse the diagram to name each geometric figure.

A. three points

Name three exact locations.

B. two lines

Name two straight paths that extend without end in opposite

directions.

C. a point shared by two lines

Name a point that is on both lines. point

D. a plane

Name three points that are on the plane, but not on a line.

, , and

Use the three points to name a plane.

plane

Identifying Line Segments and RaysUse the diagram to give a possible name to each figure.

A. two different line segments

Use the endpoints to name two different line segments.

B. six different names for rays

Use an endpoint first and then another point on the ray to name

six rays.

C. another name for ray QR

What is the endpoint? What is another point on the ray?

What is another name for ray QR?

Q

RS

G

IJ

HK

L

Vocabulary

pointlineplaneline segmentray

Copyright © by Holt, Rinehart and Winston. 156 Holt MathematicsAll rights reserved.

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Ready to Go On? Skills InterventionMeasuring and Classifying Angles8-2

LESSON

An angle is formed by two rays with a common endpoint, calledthe vertex. Angles are measured in degrees. A right anglemeasures exactly 90° and a straight angle measures exactly180°. An acute angle measures less than 90° and an obtuseangle measures more than 90° and less than 180°.

Measuring an Angle with a ProtractorUse a protractor to measure the angle. Tell what type of angle it is.

• Where will you place the center point of the protractor?

• Which degree mark on the protractor do you want ray

BC to pass through?

• Using the scale that starts with 0° along ray BC, read

the measure where ray BA crosses.

• m�ABC �

• Is m�ABC greater than, less than, or equal to 90°?

So the angle is a(n) angle.

Drawing an Angle with a ProtractorUse a protractor to draw an angle that measures 120°.

• Draw a ray near the bottom of the space provided to the right.

• Where will you place the center point of the

protractor?

• On the protractor, which degree mark do you want

the ray to pass through?

• At which degree mark will you make a mark above the scale on

the protractor?

• Where will you draw the second ray?

Vocabulary

anglevertexright anglestraight angleacute angleobtuse angle

A

B C

There are 360° in a circle. You can use that fact to help solve someproblems.

What is the measure of the angle formed by the minute and hour hands of a clock at 1:30?

Understand the Problem

1. How many degrees is it all the way around the clock face?

Make a Plan

2. Break the angle you want to find into parts. What fraction of a whole circle is �BOC? How could you use that fractionto find m�BOC?

3. If you knew m�AOB, how could you find m�DOB? Whatfraction of a whole circle would that angle be?

4. What fraction of a whole circle is �AOB? How could you usethat fraction to find m�AOB?

Solve

5. What is m�BOC? Write the measure on the diagram.

6. What is m�DOB? Write the measure on the diagram.

7. What is the measure of the angle formed by the hands at 1:30?

Check

8. Show that your answer makes sense.

Solve

9. What size angle do the hands form at 10:15? Hint: The hour

hand is �14

� of the way from 10 to 11.

Ready to Go On? Problem Solving InterventionMeasuring and Classifying Angles8-2

LESSON


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12A

DB

O

C

39

10

8 4

2

11 1

7 56


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Ready to Go On? Skills InterventionAngle Relationships8-3

LESSON

Congruent angles are angles that have the same measure.Vertical angles are formed opposite of each other when two linesintersect. Adjacent angles are side by side and have a commonvertex and ray. Complementary angles are two angles whosemeasures have a sum of 90° and supplementary angles are twoangles whose measures have a sum of 180°.

Identifying Types of Angle PairsIdentify the type of each angle pair shown.

A. Is �3 opposite �4?

Are both angles formed by two intersecting lines?

What type of angles are �3 and �4?

B. Are angles 5 and 6 side by side?

Do they have a common vertex and ray?

What type of angles are �5 and �6?

Identifying an Unknown Angle MeasureFind each unknown angle measure.

A. The angles are complementary.

45° � A � What is the sum of the measures?

� � Subtract 45° from both sides.

A � Solve for A.

m�A �

B. The angles are supplementary.

35° � B � What is the sum of the measures?

� � Subtract 35° from both sides.

B � Solve for B.

m�B �

B 35°

A45°

5 6

3 4

Vocabulary

congruentvertical anglesadjacent anglescomplementary

anglessupplementary

angles

Ready to Go On? Skills InterventionClassifying Lines8-4

LESSON


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Parallel lines are lines in the same plane that never intersect.Perpendicular lines intersect to form 90° angles, or right angles.Skew lines are lines that lie in different planes; they are neitherparallel nor intersecting.

Classifying Pairs of LinesClassify each pair of lines.

A. B.

Are the lines in the same plane? Are the lines in the same plane?

Are the lines parallel? Do the lines intersect?

Do the lines intersect? The lines are .

The lines are .

C. D.

Do the lines cross at one common Do the lines intersect to form

point? right angles?

The lines are . The lines are .

Map ApplicationMain Street and Grand Street each run North-South.What type of line relationship does this represent?

Are the streets in the same plane?

Do the streets intersect?

The streets represent lines.

Ma

in S

treet

Gra

nd

Stre

et

Vocabulary

parallel linesperpendicular linesskew lines


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Ready to Go On? Quiz8A

SECTION

8-1 Building Blocks of GeometryUse the diagram to name each geometric figure.

1. three points

2. two lines

3. a point shared by two lines

4. a plane

5. two line segments

6. two rays

8-2 Measuring and Classifying AnglesUse a protractor to measure each angle. Then classify eachangle as acute, right, obtuse, or straight.

7. 8.

9. 10.

11. The steepest slide at the new playground has a top angle of60°. Draw an angle with this measure.

F

E

G

H

8-3 Angle RelationshipsFind each unknown angle measure.

12. 13.

14. 15.

8-4 Classifying LinesClassify each pair of lines.

16. 17.

18. 19.

u20° 20°t 24°

s

50°30°

r


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Ready to Go On? Quiz continued

8ASECTION


Students in Mrs. Zirellis’ class built towers from rolling newspaperand then taping it together. In order to make a sturdy tower thesupports were organized into angles. The graphic below is a sketchof one of the taller towers.

Measuring and Classifying Angles

Use a protractor to measure each angle. Then classify each angleas acute, right, or obtuse.

1. �ABC

2. �DEF

3. �GHI

Angle Relationships

Find each unknown angle measure.

4. �J �

5. �K �

6. �L �

7. �M �

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Ready to Go On? EnrichmentPaper Tower8A

SECTION

90° 40° 55° 50° 45° 65° 50°

L

J

E

F

D

K

A

B

C

M

H

G

I


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Ready to Go On? Skills InterventionTriangles8-5

LESSON

Triangles can be classified by the measures of their angles. Anacute triangle has only acute angles. An obtuse triangle has oneobtuse angle. A right triangle has one right angle. The sum of themeasures of the angles in any triangle is 180°.

Sports ApplicationHiking paths connecting lakes J, K, and L form a triangle. Themeasure of �J is 38°, and the measure of �K is 44°. Classify the triangle.

To classify the triangle, find the measure of �L.

L � 180° � ( � ) What will you subtract from 180°?

L � 180° �

L �

The measure of �L is . Does �JKL have an obtuse angle?

So the hiking paths form a(n) triangle.

Using Properties of Angles to Label TrianglesUse the diagram to find the measure of �GIJ.

What type of angles are �GIJ and �FIH?

So m�GIJ m�FIH.

m�FIH � 180° � ( � )

� 180° �

�

m�FIH � so, m�GIJ � . F H

G

I J

20° 25°

Vocabulary

acute triangleobtuse triangleright triangle

Acute triangle Obtuse triangle Right triangle


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Ready to Go On? Skills InterventionQuadrilaterals8-6

LESSON

A quadrilateral is a plane figure with four sides and four angles.There are five special types of quadrilaterals.parallelogram: opposite sides are parallel and congruent, andopposite angles are congruentrectangle: parallelogram with four right anglesrhombus: parallelogram with four congruent sidessquare: rectangle with four congruent sidestrapezoid: exactly two parallel sides, may be two right angles

Naming QuadrilateralsGive the most descriptive name for each figure.

A. Does the figure have four sides and four angles?

Are exactly two of the sides parallel?

is the most exact name.

B. Is this figure a plane figure?

Does the figure have only four sides and four angles?

The figure is a quadrilateral.

C. Does the figure have four sides and four angles?

Does the figure have four right angles?

is the most exact name.

Classifying QuadrilateralsComplete the statement.

A. A rhombus can also be called a and .

Does a rhombus have opposite sides that are parallel and congruent?

So a rhombus can be called a .

Is a rhombus a plane figure with four sides and four angles?

So a rhombus can be called a .

B. A rhombus with four right angles can also be called a .

Does a square have four congruent sides and four right angles?

So a rhombus with four right angles can also be called a .

?

??

Vocabulary

quadrilateralparallelogramrectanglerhombussquaretrapezoid

Sometimes you can solve a problem even though you may think atfirst that there is not enough information.

A rhombus and an equilateral triangle are adjacent, forming a trapezoid. What fraction of the perimeter of the trapezoid is the perimeter of the rhombus?


1. Draw a rhombus and triangle as described in the problem.

2. What two quantities are you supposed to compare?

Make a Plan

3. What do you know about length of the sides of a rhombus? Of an equilateral triangle?

4. Why must the sides of the triangle be the same length as thesides of the rhombus?

5. Mark your diagram to show which lengths are equal.

Solve

6. If the length of each side of the rhombus is 1, what is the perimeter of the rhombus? The perimeter of thetrapezoid?

7. What fraction of the perimeter of the trapezoid is the perimeterof the rhombus?

Check

8. Does your diagram match what the problem states? Is yourreasoning correct?

Ready to Go On? Problem Solving InterventionQuadrilaterals8-6

LESSON


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Ready to Go On? Skills InterventionPolygons8-7

LESSON

A polygon is a closed plane figure formed by three or more linesegments. A regular polygon is a polygon with all sidescongruent and all angles congruent.

Identifying PolygonsName each polygon and tell whether it appears to be regular ornot regular.

A. How many sides are there?

How many angles are there?

The polygon is a(n) .

Do the sides and angles appear to be congruent?

The polygon appears to be .

B. How many sides are there?





C. How many sides are there?





Home Economics ApplicationJessica made a cake in the shape of a regular hexagon for herbrother’s sixth birthday. What is the measure of each angle ofthe hexagon?

How many sides does a hexagon have?

How many triangles are there inside a hexagon?

What is the sum of the interior angles in a hexagon? � 180° �

So the measure of each angle is � or .

Vocabulary

polygonregular polygon

Extending Geometric PatternsIdentify a possible pattern. Use the pattern to draw the next figure.

What might the pattern be?

So the next figure might be a .

Completing Geometric PatternsIdentify a possible pattern. Use the pattern to draw the missingfigure.


So the missing figure might be a .

Art ApplicationShannon is drawing a picture. Identify a pattern that Shannon isusing and draw what the next shape will probably be.


So the missing figure might be a

.

Ready to Go On? Skills InterventionGeometric Patterns8-8

LESSON


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Ready to Go On? Quiz8B

SECTION

8-5 TrianglesUse the diagram for problems 1 and 2.

1. Find m �HFD.

2. Classify triangle HGE by its angles and by its sides.

If the angles can form a triangle, classify it as acute, obtuse, or right.

3. 98°, 48°, 34° 4. 52°, 38°, 90°

5. 73°, 57°, 60° 6. 75°, 65°, 40°

8-6 QuadrilateralsGive the most descriptive name for each figure.

7. 8.

9. 10.

11. One angle of a rhombus is 73°. What is the measure of the opposite angle?

12. The perimeter of a square is 168 centimeters. What is the length of one side of the square?

60° 40°

F

D

E G

H

3.5 m

6 m

5 m

1.5 m


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8BSECTION

8-7 PolygonsName each polygon and tell whether it appears to be regularor irregular.

13. 14.

15. 16.

8-8 Geometric PatternsIdentify a possible pattern. Use the pattern to draw themissing figure.

17. 18.

19. 20.

??

??


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Ready to Go On? EnrichmentCustom Kites8B

SECTION

Here are diagrams of some kites at a local kite festival.

Quadrilaterals

Give all the names for each polygon.

1. 2.

3. 4.

Polygons

Name each polygon and tell whether it appears to be regularor irregular.

5. 6.

7. 8.


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Identifying Congruent FiguresDecide whether the figures in each pair are congruent.

A. Do these figures have the same shape

and size?

These figures are .

B. Are both of these figures squares?

Do these figures have the same shape

and size?

These figures are .

C. Are both of these figures triangles?

Do these figures have the same shape

and size?

These figures are .

Consumer ApplicationShauna needs a tablecloth that is congruent to the top of the table.Which tablecloth should she buy?

Table Tablecloth A Tablecloth B

Are both tablecloths the same shape as the table?

Are both tablecloths the same size as the table?

Which tablecloth is the same size and shape as the table?

is congruent to the table.

8 ft

4 ft

4 ft2 ft

8 ft

4 ft

6 in.

6 in.

10 in.

8 in.10 in.

8 in.

Ready to Go On? Skills InterventionCongruence8-9

LESSON


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If you know that two triangles are congruent, you may be able touse that information to prove other things.

In the diagram, �BCE � �ACD.Show that B�D� � A�E�.


1. Use a colored marker to outline the two congruenttriangles BCE and ACD.

Make a Plan

2. Mark the sides of the triangles that are congruent.

3. What two line segments make up side �A�C�? side �B�C�?

Solve

4. Fill in each blank with the correct distance.

A�C� � A�E� � B�C� � �

A�C� � E�C� �

A�C� � E�C� � B�C� � D�C� �

5. Use exercise 4 to explain how you know that A�E� � B�D�.

Check

6. Trace �BCE and �ACD to make sure you identified correspondingsides correctly.

Solve

7. Which triangle has the larger perimeter, �BCE or �ACD? Explain.

8. If B�C� � 6 cm and A�E� � 4 cm, what is the length of D�C�?

Ready to Go On? Problem Solving InterventionCongruence8-9

LESSON

B D C

E

A


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A rigid transformation moves a figure without changing its size orshape. A translation is the movement of a figure along a straightline. A rotation is the movement of a figure around a point. Whena figure flips over a line, creating a mirror image, it is called areflection. The line the figure is flipped over is called the line ofreflection.

Identifying TransformationsTell whether each is a translation, rotation, or reflection.

A. How did the figure move?

The figure is a .

B. How did the figure move?

The figure is a .

C. How did the figure move?

The figure is a .

Drawing TransformationsDraw each transfomation.

A. Draw a vertical reflection.

What will you flip the figure over?

Draw the figure in its new location.

B. Draw a 90° clockwise rotation about the point.

Where will you place your pencil?

How far will you rotate the figure?

Draw the figure in its new location.

Ready to Go On? Skills InterventionTransformations8-10

LESSON

Vocabulary

transformationtranslationrotation

reflectionline of reflection

MSM07C1_RTGO_ch08_156-178_B 6/18/06 12:20 PM Page 174 (Black plate)


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A figure has line symmetry if it can be folded or reflected so thatthe two parts of the figure match, or are congruent. The line ofreflection is called the line of symmetry.

Identifying Lines of SymmetryDetermine whether the dashed line appears to be a line of symmetry.

Are the two parts of the figure congruent?

Do they appear to match exactly when folded or reflected across

the line?

The line to be a line of symmetry.

Finding Multiple Lines of Symmetry

A. Find all the lines of symmetry in the regular polygon.

Trace the figure and cut it out.

Fold the figure in half in different ways.

How many different ways can you fold the figure in half?

So there are lines of symmetry.

B. Find all the lines of symmetry in the object.

Trace the figure and cut it out.

Fold the figure in half in different ways.

How many different ways can you fold the figure in half?

So there are lines of symmetry.

Visualizing Symmetric FiguresDraw the cut-out figure Then reflect that figure as it would look unfolded. Draw a congruent figure. across the fold line.

Ready to Go On? Skills InterventionLine Symmetry8-11

LESSON

Vocabulary

line of symmetryline symmetry

8-9 CongruenceDecide whether the figures in each pair are congruent. If not, explain.

1. 2.

3. Jon needs a cover for his pool. Which cover will fit?

8-10 TransformationsTell whether each is a translation, rotation, or reflection.

4. 5.

6. 7.

6 m

Pool6 m5

m

6 m

5 m

5 m3

m

5 m

ACB


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Ready to Go On? Quiz8C

SECTION


Draw and name each transformation.

8. 9.

8-11 Line SymmetryDetermine whether each dashed line appears to be a lineof symmetry.

10. 11.

12. 13.

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8CSECTION

At summer camp, Jill took a jewelry-making class. She wanted to make a belt bucklesimilar to the ones she saw in a design book. The book instructed her to look at theplacement of objects and how identical objects are put into place.

Jill starts with the jeweled belt buckle.

1. The first step to assemble the belt buckle is to transform the star in the top left corner to the other three corners as shown below. Is this a translation, rotation, or reflection?

2. The next step is to transform the top center design to the lower position as shown. Is this a translation, rotation, or reflection?

3. Lastly, attach a string of stones from the center circle. Repeat several strings around the circle. Are these translations, rotations, or reflections?

4. Complete the rest of the bracelet using rotation.

5. Draw your own piece of jewelry using transformations. Labeleach transformation type.


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Ready to Go On? EnrichmentTransforming Jewelry8C

SECTION

lesson ready to go on? skills intervention building …€¦ · · 2011-09-11ready to go on?...

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