Parallel Lines cut by a Transversal - PowerPoint Presentation

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Uploaded On 2019-06-19

Parallel Lines cut by a Transversal - PPT Presentation

Recall that parallel lines are two coplanar lines that never intersect AB CD A transversal is a line segment or ray that intersects two or more lines at different distinct points ID: 759183

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Slide1

Parallel Lines cut by a Transversal

Slide2

Recall that parallel lines are two coplanar lines that never intersect.AB || CD

Slide3

A transversal is a line, segment or ray that intersects two or more lines at different distinct points.

EXAMPLES: In each case of examples , t is the transversal

NON-EXAMPLES: In the non-example case, t is NOT a transversal

Same point>>>

Only one line

Do not intersect

Slide4

When parallel lines are cut by a transversal,

Several special types of pairs of

angles are formed. They are named based on the angles positions relative to the parallel lines and the transversal.

Slide5

When parallel lines are cut by a transversal,

eight angles are formed

Slide6

Each parallel line intersects with the transversal to create

two sets of 4 angles.

SET 1

SET 2

Slide7

The top set of a

ngles (1,2,3,and 4) formed

have

exactly the same measures as the bottom set ( 5,6,7, and 8) when m and n are parallel because they are being cut by the same line (t) at the same angle.

Slide8

The top set of a

ngles (1,2,3,and 4) could be cut and placed on top of the bottom set (5,6,7,and 8)

Slide9

Four sets of

corresponding angles are formed.Corresponding angles are angles that lie in the same position when two lines are cut by a transversal.1 corresponds to 5 because they are both in the top left position.

Slide10

Four sets of corresponding angles are formed.

Corresponding angles are angles that lie in the same position when two lines are cut by a transversal.2 corresponds to 6 because they both lie in the top right corner of the sets.

Slide11

Four sets of corresponding angles are formed.

Corresponding angles are angles that lie in the same position when two lines are cut by a transversal.3 corresponds to 7 because they both lie in the bottom left corner of the sets

Slide12

Four sets of corresponding angles are formed.

Corresponding angles are angles that lie in the same position when two lines are cut by a transversal.4 corresponds to 8 because they both lie in the bottom right corner of the sets.

Slide13

Corresponding Angles TheoremIf two parallel lines are cut by a transversal, corresponding angles are congruent

Therefore we know

<1 <5<2 <6<3 <7 <4 <8

|| n

Slide14

Find the measure of angle 2

2x + 100 corresponds to 5x + 55

Therefore 2x+100 = 5x + 55

45 = 3x

15 = x 5x +55 =5(15)+55 = 130o

5x +55 and <2 are linear pair so5x + 55 + <2 = 180o 130 + <2 = 180o <2 = 50o

130o

Bottom right - Corresponding angles

|| n

Slide15

When parallel lines are cut by a transversal,

eight angles are formed

The parallel lines, m and n, cut two areas in the planecalled the interior……

|| n

Slide16

When parallel lines are cut by a transversal,

eight angles are formed

The parallel lines, m and n, cut two areas in the planecalled the interior and the exterior

|| n

Slide17

The angles between the parallel lines ( m and n) are

INTERIOR ANGLES

Angles 3 , 4, 5 , and 6 are interior angles.

|| n

Slide18

The angles outside the parallel lines ( m and n ) are

EXTERIOR ANGLES

Angles 1,2, 7, and 8 are exterior angles

|| n

Slide19

The transversal, t , cuts the plane into two regions.

|| n

Slide20

The transversal, t , cuts the plane into two regions.

Angles 2, 4, 6, and 8 lie on the same side of the transversal.

|| n

Slide21

The transversal, t , cuts the plane into two regions.

Angles 1, 3, 5, and 7 are on the same side of the transversal

|| n

Slide22

The transversal, t , cuts the plane into two regions.

Angles 3 and 6 are on “opposite” sides or “alternate” sides of the transversal.

|| n

Slide23

Opposite angles

fall on alternate sides of the transversalOther examples of pairs of opposite angles are 3 and 6

|| n

5 and 4

7 and 2

8 and 1

Slide24

Name two pairs of

alternate interior angles

<c and <e

<d and <f

m || n

Slide25

Name two pair of

same- side interior angles

<c and <f

<d and <e

m || n

Slide26

Name two pairs of

alternate exterior angles

<a and <g

<b and <h

m || n

Slide27

Name two pairs of same-side

exterior angles

<a and <h

<b and <g

m || n

Slide28

Name four pairs of corresponding angles

<c and <g

<d and <h

<b and <f

<a and <e

|| n

Slide29

Name four pairs of vertical angles

<e and <g

<f and <h

<b and <d

<a and <c

|| n

Slide30

Name eight pairs of supplementary angles

<c and <d

<d and <a

<b and <c

<a and <b

<e and <f

<f and <g

<g and <h

<h and <e

|| n

Slide31

Alternate Interior Angles Theorem

Slide32

GIVEN : p || q

Prove : <1

Slide33

Slide34

Slide35

Slide36

Slide37

Same-Side Exterior Angles Theorem

If two parallel lines are cut by a transversal, then same-side exterior angles are supplementary.

< 1 + <3 = 180o

a || b

Slide38

GIVEN:

l || mPROVE: m<4 +m<5 = 180o

STATEMENTSREASONS1. l || m1. Given2. Corresponding <‘s Theorem3. = 180O3. Defn. of linear pair4. = 180O4. Substitution Property

STATEMENTSREASONS1. l || m1. Given2. Corresponding <‘s Theorem3. Defn. of linear pair4. Substitution Property

Same-Side Exterior Angles Theorem

Slide39

When 2 || lines are cut by a transversal,

Congruent

Pairs of angles

Vertical Angles

Corresponding Angles

Alternate Exterior Angles

Alternate Interior Angles

Slide40

When 2 || lines are cut by a transversal,

Supplementary

Pairs of angles

Linear Pair

Same-Side Interior Angles

Same-Side Exterior Angles

Slide41

HOMEWORK

Complete the problems

on the following pages

Slide42

Given: a || b and c || d < 9 = 81oFind the measures of all the angles. Justify your reasoning using pairs of special angles.

Slide43

Given: a || b and c || d < 9 = 81oFind the measures of all the angles. Justify your reasoning using pairs of special angles.

81o

99o

Slide44

17) Solve for x and find the measures of the angles

Slide45

17) Solve for x and find the measures of the angles

X+75

+ x+125 = 180 2x + 200 = 180 2x = -20 x = -10

-10 + 75 = 65

-10 + 125 = 115

Define the relationship -

same-side interior angles so they are supplementary ----SUM IS 180o

3.) Substitute the value of x in the expressions and find the measures of the angles to answer the question

2.) Write equation and solve for x

Slide46

18) Solve for x and find the measures of the angles

Slide47

18) Solve for x and find the measures of the angles

Define the relationship

- corresponding angles so they are congruent – measures are equal.

12x + 3 = 11x + 9 x = 6

2.) Write equation and solve for x

3.) Substitute the value of x in the

expressions and find the

measures of the angles

to answer the question

12x + 3 = 12(6)+ 3 = 75o11x + 9 = 11(6) + 9 = 75o

Slide48

19) Solve for x and find the measures of the angles

Slide49

19) Solve for x and find the measures of the angles

Define the relationship

- ( 15x -5)o and 125o are a linear pair so their sum is 180o . (7y +27) o and 125o are alternate exterior angles so they are congruent.

2.) Write equation and solve for x

( 15x -5)o + 125o= 180o (7y +27) o = 125o 15x +120 = 180o 7y = 98o 15x = 60o y = 14o x = 4 15x -25 = 15(4) -25 = 35 7y +27 = 125o 7(14) +27 = 125o 125o = 125o

3.) Substitute the value of x in the expressions and find the measures of the angles to answer the question

Slide50

Solve for x and find the measures of the angles

AB || CD

Slide51

Solve for x and find the measures of the angles

AB || CD

1.) Alternate interior angles are congruent

2.) 120 = 3x

40 = x

3.) 3(x) = 3(40) = 120

Slide52

Solve for x and y.

Then find the measures of the angles

Slide53

Solve for x and y.

Then find the measures of the angles

1.) (6x +y) and ( x + 5y) are corresponding angles so they are congruent. 4x and ( 6x + y ) are a linear pair so they are supplementary and have a sum of 180.

2.) 6x +y = x + 5y and 4x + 6x + Y = 180 5X = 4Y 10X +y = 180 y = 180 – 10x

Since there are 2 variables in both equations, you have a system. Y = 180 – 10x 5x = 4y Solve by substitution 5x = 4 ( 180 – 10x) 5(16) = 4y 5x = 720 – 40x 80 = 4y 45x =720 20 = y x = 16

3.) 4x = 4(16) = 64O 6x + Y = 6 (16) + 20= 116 O X+5Y = 16 +5(20) = 116O

Slide54

22.) Solve for x and y.

Then find the measures of the angles

Slide55

22.) Solve for x and y.

Then find the measures of the angles

1.) Same Side Interior Angles Supplementary 3y +5 + 5y+15 = 180 Linear Pair =180 5y +15 +2x+5 = 1802.) Simplify and solve to find the variables 3y +5 + 5y+15 = 180 5y +15 +2x+5 = 180 8y +20 = 180 5y +2X +20 = 180 8Y = 160 5Y + 2X = 160 Y = 20 5(20) +2X = 160 100 +2X = 160 2X = 60 X = 30

17X -70 = 17(30)-70 = 4403Y +5 = 3(20) +5 = 60 +5 = 655Y+15 = 5(20) +15 = 1152X+5 = 2(30) + 5 65

VERTICAL ANGLES

Slide56

23) Solve for x and find the measures of the angles

Slide57

23) Solve for x and find the measures of the angles

Alternate exterior angles are =

X2 -2X -5 = 2X2 – 7X -19 0 = X2 – 5x -14 0 = ( x -7)(x +2) so…… x = 7 or x = -2 If x = 7…………….. If x = -2…………… X2 -2X -5 = (7)2 -2(7) -5 = 30 X2 -2X -5 = (-2)2 -2(-2) -5= 32X2 – 7X -19 =2(7)2 –7(7) -19 =30 2X2 – 7X -19= 2(-2)2 –7(-2) -19 = 3

Slide58

Solve for x and y.

Then find the measures of the angles.

c||d

and

a|| b

Slide59

Solve for x and y.

Then find the measures of the angles. c||d and a|| b

2x+13 = 3x -24 37 = x2(37)+13 = 873(37) -24 = 87

3y+24 + 87 = 180 3y + 111 = 180 3y = 69 y = 233y + 24 = 3(23) + 24 = 934y – 5 = 4( 23) – 5 = 87

Alternate Interior Angles =