chapter 3 danny ramsey, ashley krieg, kyle jacobs, and chris runion

Chapter 3Chapter 3

Danny Ramsey, Ashley Krieg, Danny Ramsey, Ashley Krieg, Kyle Jacobs, and Chris RunionKyle Jacobs, and Chris Runion

What is Ch. 3 About?What is Ch. 3 About?

It is about lines and angles.It is about lines and angles.We learnedWe learned::

the properties of parallel and perpendicular the properties of parallel and perpendicular lines.lines.

Six different ways to prove lines are Six different ways to prove lines are parallel.parallel.

How to write an equation of a line with the How to write an equation of a line with the given characteristics given characteristics

3.13.1

Lines and AnglesLines and Angles

3.1 Vocabulary3.1 Vocabulary

Parallel lines: two lines that are coplanar Parallel lines: two lines that are coplanar and never intersectand never intersect

Skew lines: two lines that are not coplanar Skew lines: two lines that are not coplanar and never intersectand never intersect

Parallel planes: two planes that never Parallel planes: two planes that never intersectintersect

Review…Review…

RU || WZRU || WZ WZ and TY are skew WZ and TY are skew

lineslines Plane RUZW and Plane RUZW and

plane STYX are plane STYX are parallel planesparallel planes

PostulatesPostulates

Parallel Postulate: If there is a line and a Parallel Postulate: If there is a line and a point not on the line, then there is exactly point not on the line, then there is exactly one line through the point parallel to the one line through the point parallel to the given linegiven line

P

l

There is exactly one line through point P parallel to l.

Postulates ContinuedPostulates Continued

Perpendicular postulate: If there is a line Perpendicular postulate: If there is a line and a point not on the line, then there is and a point not on the line, then there is exactly one line through the point exactly one line through the point perpendicular to the given line.perpendicular to the given line.

P

l

There is exactly one line through P perpendicular to l.

Construction ActivityConstruction Activity

Perpendicular linesPerpendicular lines

Draw a line (Draw a line (ll) and a Point (P) off of the line. ) and a Point (P) off of the line. Put point of compass at P and open wide Put point of compass at P and open wide enough to intersect enough to intersect ll twice. Label those twice. Label those intersections A and B. Using the same intersections A and B. Using the same

radius, draw an arc from A and B. Label radius, draw an arc from A and B. Label the intersection Q. Use a straightedge to the intersection Q. Use a straightedge to

draw PQ. PQ draw PQ. PQ ll

More Vocabulary!More Vocabulary! Transversal: the line that intersects two or more Transversal: the line that intersects two or more

coplanar lines at different pointscoplanar lines at different points Corresponding Angles: angles that occupy Corresponding Angles: angles that occupy

corresponding positionscorresponding positions Alternate Exterior Angles: two angles that are Alternate Exterior Angles: two angles that are

outside the two lines on opposite sides of the outside the two lines on opposite sides of the transversaltransversal

Alternate Interior Angles: two angles between Alternate Interior Angles: two angles between the two lines on opposite sides of the transversalthe two lines on opposite sides of the transversal

Consecutive Interior Angles: two angles that lie Consecutive Interior Angles: two angles that lie between the two lines on the same side of the between the two lines on the same side of the transversaltransversal

VOCAB PICTURESVOCAB PICTURESTransversal is RED

1 2

34

5 6

78

Corresponding Angles: 1 & 5

Alternate Exterior Angles: 1 & 8

Alternate Interior Angles: 3 & 6Consecutive Interior Angles: 3 & 5

3.23.2

Proof and Perpendicular LinesProof and Perpendicular Lines

3.2 Vocabulary3.2 Vocabulary

Flow Proof: uses arrows and boxes to Flow Proof: uses arrows and boxes to show the logical flowshow the logical flowExampleExample

3.2 Theorems3.2 Theorems

If two lines intersect to form a linear pair of If two lines intersect to form a linear pair of congruent angles, then the lines are congruent angles, then the lines are perpendicular.perpendicular.

If two sides of two adjacent acute angels If two sides of two adjacent acute angels are perpendicular, then the angles are are perpendicular, then the angles are complementary.complementary.

If two lines are perpendicular, then they If two lines are perpendicular, then they intersect to form four right angles.intersect to form four right angles.

3.33.3

Parallel lines and TransversalsParallel lines and Transversals

Corresponding Angles PostulateCorresponding Angles Postulate

If two parallel lines are cut by a If two parallel lines are cut by a transversal, then the pairs of transversal, then the pairs of corresponding angles are congruentcorresponding angles are congruent

1

2

<1 = <2


Alternate Interior Alternate Interior Angles: if two Angles: if two parallel lines are parallel lines are cut by a cut by a transversal, then transversal, then the pairs of the pairs of alternate interior alternate interior angles are angles are congruentcongruent

34

<3 = <4


Consecutive Consecutive Interior Angles: if Interior Angles: if two parallel lines two parallel lines are cut by a are cut by a transversal, then transversal, then the pairs of the pairs of consecutive interior consecutive interior angels are angels are supplementary.supplementary.

5

6

M<5 + M<6 = 180˚


Alternate Exterior Alternate Exterior Angles: if two Angles: if two parallel lines are parallel lines are cut by a cut by a transversal, then transversal, then the pairs of the pairs of alternate exterior alternate exterior angles are angles are congruentcongruent

7

8

<7 = <8


Perpendicular Perpendicular Transversal: if a Transversal: if a transversal is transversal is perpendicular to perpendicular to one of two parallel one of two parallel lines, then it is lines, then it is perpendicular to perpendicular to the otherthe other

j

h

k

J K

3.43.4

Proving Parallel LinesProving Parallel Lines

POSTULATEPOSTULATE

Corresponding angles converse: if two Corresponding angles converse: if two lines are cut by a transversal so that lines are cut by a transversal so that corresponding angles are congruent, then corresponding angles are congruent, then the pairs of alternate interior angles are the pairs of alternate interior angles are congruentcongruent

j

k

J || k

Theorems about TransversalsTheorems about Transversals

Alternate Interior Alternate Interior Angles Converse: if Angles Converse: if two lines are cut by a two lines are cut by a transversal so that transversal so that alternate interior alternate interior angles are congruent, angles are congruent, then the lines are then the lines are parallelparallel

3

1

j

k

If <1 = <3, then j || k


Consecutive Interior Consecutive Interior Angles Converse: if Angles Converse: if two lines are cut by a two lines are cut by a transversal so that transversal so that consecutive interior consecutive interior angles are angles are supplementary, then supplementary, then the lines are parallelthe lines are parallel

2

1

If m<1 + m<2 = 180°, j || k

j

k


Alternate Exterior Alternate Exterior Angles Converse: if Angles Converse: if two lines are cut by a two lines are cut by a transversal sot that transversal sot that alternate exterior alternate exterior angles are congruent, angles are congruent, then the lines are then the lines are parallel.parallel.

4

5

If <4 = <5, then j || k

j

k

3.53.5

Using Properties of Parallel LinesUsing Properties of Parallel Lines

TheoremsTheorems

If two lines are If two lines are parallel to the same parallel to the same line, then they are line, then they are parallel to each other.parallel to each other.

In a plane, if two lines In a plane, if two lines are perpendicular to are perpendicular to the same line, then the same line, then they are parallel to they are parallel to each othereach other

pq

r

If p || q and q || r, p || r


Copying an AngleCopying an Angle Draw an acute angle with the vertex ADraw an acute angle with the vertex A Below the angle, draw a line using a straight edge put a Below the angle, draw a line using a straight edge put a

point on the line and label it Dpoint on the line and label it D Using a compass, put the point on A and open wide Using a compass, put the point on A and open wide

enough to intersect both rays. Label the intersections B enough to intersect both rays. Label the intersections B and Cand C

Using the same radius on the compass, draw an arc with Using the same radius on the compass, draw an arc with the center D, label the intersection Ethe center D, label the intersection E

Draw an arc with the radius BC and center E, label the Draw an arc with the radius BC and center E, label the intersection Fintersection F

Draw DF. <EDF = <BAC Draw DF. <EDF = <BAC


Parallel LinesParallel Lines Draw line M, using a straight edge, and point P off of the Draw line M, using a straight edge, and point P off of the

line.line. Draw points Q and R on line M. Draw PQDraw points Q and R on line M. Draw PQ Draw an arc with the center at Q so it crosses QP and Draw an arc with the center at Q so it crosses QP and

QRQR Now copy <PQR, as shown in the previous construction Now copy <PQR, as shown in the previous construction

activity, on QP. Be sure the angles are corresponding,activity, on QP. Be sure the angles are corresponding, Label the new angle <TPSLabel the new angle <TPS Draw PS. Since <TPS and <PQR are congruent Draw PS. Since <TPS and <PQR are congruent

corresponding angles, PS || QRcorresponding angles, PS || QR

3.63.6

Parallel Lines in the Coordinate Parallel Lines in the Coordinate PlanePlane

YOU MUST KNOW THIS!!!YOU MUST KNOW THIS!!!

RISE

RUN

SLOPE =Y2 – Y1

X2 – X1

= M

Now, the pictureNow, the picturey

x

(X1 , Y1)

(X2 , Y1)

Y2 – Y1

RISE

X2 – X1

RUN

Slope of Parallel Lines PostulateSlope of Parallel Lines Postulate

In a coordinate plane, In a coordinate plane, two nonvertical lines two nonvertical lines are parallel is and are parallel is and only if they have the only if they have the same slope. Any two same slope. Any two vertical lines are vertical lines are parallel.parallel.

Slope = -1

SLOPESSLOPES

Lines that have the same slope are Lines that have the same slope are parallel. Y = 2x + 3 ; Y = 2x – 6parallel. Y = 2x + 3 ; Y = 2x – 6

Lines that are perpendicular have opposite Lines that are perpendicular have opposite reciprocal slopes. Y = -2x + 3 ; Y = 1/2x -9reciprocal slopes. Y = -2x + 3 ; Y = 1/2x -9

3.73.7

Perpendicular lines in the Perpendicular lines in the coordinate planecoordinate plane

Slopes of Perpendicular LinesSlopes of Perpendicular Lines

In a coordinate plane, In a coordinate plane, two nonvertical lines two nonvertical lines are perpendicular if are perpendicular if and only if the product and only if the product of their slopes is -1. of their slopes is -1. Vertical and Vertical and horizontal lines are horizontal lines are perpendicular.perpendicular.

Product of Slopes: 2 ( - ½ ) = -1

But Wait!!!But Wait!!!

When Will I Ever Use This???When Will I Ever Use This???

SailingSailing There are three basic sailing maneuvers - There are three basic sailing maneuvers -

sailing into the wind, sailing across the wind, sailing into the wind, sailing across the wind, and sailing with the wind. These three and sailing with the wind. These three maneuvers allow a sailboat to travel in maneuvers allow a sailboat to travel in almost any direction. A boat that is sailing almost any direction. A boat that is sailing into (or against) the wind is actually sailing into (or against) the wind is actually sailing at an angle of about 45° to the direction of at an angle of about 45° to the direction of the wind. A sailboat that is sailing into the the wind. A sailboat that is sailing into the wind must follow a zigzag course called wind must follow a zigzag course called tackingtacking in order to avoid sailing directly into in order to avoid sailing directly into the wind. When a boat is pointed directly the wind. When a boat is pointed directly into the wind, the sails are rendered useless into the wind, the sails are rendered useless and the boat loses its ability to move. A boat and the boat loses its ability to move. A boat can reach maximum speed by sailing can reach maximum speed by sailing across the wind or across the wind or reaching.reaching. In this In this situation, the wind direction is perpendicular situation, the wind direction is perpendicular to the side of the boat. The third sailing to the side of the boat. The third sailing technique is called sailing with the wind or technique is called sailing with the wind or running.running. Here, the sail is almost at right Here, the sail is almost at right angles with the boat and the wind literally angles with the boat and the wind literally pushes the boat from the stern pushes the boat from the stern

Graphic Artists Graphic Artists Graphic artists are creative, Graphic artists are creative,

analytical, and detail-oriented. analytical, and detail-oriented. It is important to be able to It is important to be able to create a visual image of an create a visual image of an idea. This talent requires idea. This talent requires strong spatial reasoning skills. strong spatial reasoning skills. The use of various types of The use of various types of graphic design software graphic design software involves an understanding of involves an understanding of geometric ideas such as geometric ideas such as scaling and transformations, scaling and transformations, and an understanding of the and an understanding of the use of percents in mixing use of percents in mixing colors.colors.

REAL WORLD PROBLEMREAL WORLD PROBLEM

90 degrees

85 degreesApple St.

Orange St.

Watermelon Ave.

Are Apple and Orange Streets Parallel?

Are there any Perpendicular Intersections?

chapter 3 danny ramsey, ashley krieg, kyle jacobs, and chris runion

Documents

coplanar lines

vocabulary parallel

point parallel

given line parallel

skew lines wz

point p parallel

transversal slide

point perpendicular