Parallel Lines cut by a Transversal

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

EXAMPLES: In each case of examples , t is the transversal A transversal is a line, segment or ray that intersects two or more lines at different distinct points. t EXAMPLES: In each case of examples , t is the transversal w t v r t n m p q NON-EXAMPLES: In the non-example case, t is NOT a transversal d c t a Only one line Same point>>> Do not intersect t b t

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.

When parallel lines are cut by a transversal, eight angles are formed

Each parallel line intersects with the transversal to create two sets of 4 angles. SET 1 SET 2

The top set of angles (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.

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

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.

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.

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

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.

Corresponding Angles Theorem If two parallel lines are cut by a transversal, corresponding angles are congruent m || n Therefore we know <1 ≅ <5 <2 ≅ <6 <3 ≅ <7 <4 ≅ <8 3 4 8 7

m || n Find the measure of angle 2 Bottom right - Corresponding angles 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 so 5x + 55 + <2 = 180o 130 + <2 = 180o <2 = 50o

m || n When parallel lines are cut by a transversal, eight angles are formed The parallel lines, m and n, cut two areas in the plane called the interior…… m || n

m || n When parallel lines are cut by a transversal, eight angles are formed The parallel lines, m and n, cut two areas in the plane called the interior and the exterior m || n

m || n The angles between the parallel lines ( m and n) are INTERIOR ANGLES Angles 3 , 4, 5 , and 6 are interior angles. m || n

m || n The angles outside the parallel lines ( m and n ) are EXTERIOR ANGLES Angles 1,2, 7, and 8 are exterior angles m || n

The transversal, t , cuts the plane into two regions. m || n

m || n The transversal, t , cuts the plane into two regions. Angles 2, 4, 6, and 8 lie on the same side of the transversal. m || n

m || n The transversal, t , cuts the plane into two regions. Angles 1, 3, 5, and 7 are on the same side of the transversal. m || n

m || n The transversal, t , cuts the plane into two regions. Angles 3 and 6 are on “opposite” sides or “alternate” sides of the transversal. m || n

m || n Opposite angles fall on alternate sides of the transversal Other examples of pairs of opposite angles are 3 and 6 8 and 1 7 and 2 5 and 4 m || n

Name two pairs of alternate interior angles m || n <c and <e <d and <f

Name two pair of same- side interior angles m || n <c and <f <d and <e

Name two pairs of alternate exterior angles m || n <a and <g <b and <h

Name two pairs of same-side exterior angles m || n <a and <h <b and <g

Name four pairs of corresponding angles m || n <a and <e <b and <f <c and <g <d and <h

Name four pairs of vertical angles m || n <a and <c <b and <d <e and <g <f and <h

Name eight pairs of supplementary angles m || n <a and <b <b and <c <c and <d <d and <a <e and <f <f and <g <g and <h <h and <e

Alternate Interior Angles Theorem

GIVEN : p || q Prove : <1 ≅ <2 p 1 2 3 q t

Same-Side Exterior Angles Theorem a || b If two parallel lines are cut by a transversal, then same-side exterior angles are supplementary. < 1 + <3 = 180o

Same-Side Exterior Angles Theorem GIVEN: l || m PROVE: m<4 +m<5 = 180o 4 5 STATEMENTS REASONS 1. l || m 1. Given <6 ≅ <4 , 2. Corresponding <‘s Theorem 3. <6 + <5 = 180O 3. Defn. of linear pair 4. <4 + <5 = 180O 4. Substitution Property

When 2 || lines are cut by a transversal, Congruent Pairs of angles Vertical Angles Corresponding Angles Alternate Exterior Angles Alternate Interior Angles

Supplementary Pairs of angles When 2 || lines are cut by a transversal, Supplementary Pairs of angles Linear Pair Same-Side Interior Angles Same-Side Exterior Angles

HOMEWORK Complete the problems on the following pages

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

Given: a || b and c || d < 9 = 81o Find the measures of all the angles. Justify your reasoning using pairs of special angles. 81o 99o 81o 99o 99o 81o 99o 81o 81o 99o 81o 99o 99o 99o 81o 81o

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

17) Solve for x and find the measures of the angles Define the relationship - same-side interior angles so they are supplementary ----SUM IS 180o X+75 + x+125 = 180 2x + 200 = 180 2x = -20 x = -10 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 -10 + 75 = 65 -10 + 125 = 115

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

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 = 75o 11x + 9 = 11(6) + 9 = 75o

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

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

Solve for x and find the measures of the angles AB || CD

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

Solve for x and y. Then find the measures of the angles

Then find the measures of the angles 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

22.) Solve for x and y. Then find the measures of the angles

Then find the measures of the angles 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 = 180 2.) 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 VERTICAL ANGLES 17X -70 = 17(30)-70 = 440 3Y +5 = 3(20) +5 = 60 +5 = 65 5Y+15 = 5(20) +15 = 115 2X+5 = 2(30) + 5 65

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

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= 3 2X2 – 7X -19 =2(7)2 –7(7) -19 =30 2X2 – 7X -19= 2(-2)2 –7(-2) -19 = 3

Solve for x and y. Then find the measures of the angles. c||d and a|| b

Then find the measures of the angles. c||d and a|| b Solve for x and y. Then find the measures of the angles. c||d and a|| b 87 93 87 Alternate Interior Angles = 87 Linear Pair have sum of 180 2x+13 = 3x -24 37 = x 2(37)+13 = 87 3(37) -24 = 87 3y+24 + 87 = 180 3y + 111 = 180 3y = 69 y = 23 3y + 24 = 3(23) + 24 = 93 4y – 5 = 4( 23) – 5 = 87

25.) Solve for all the variables. Then find the measures of the angles.

25.) Solve for all the variables. Then find the measures of the angles. ONE VARIABLE! Same side interior angles are congruent A + 30 = 60 A = 30 60

Corresponding angles = 60 30 +2b = 60 2b = 30 b = 13

Vertical angles 25.) Solve for all the variables. Then find the measures of the angles. Vertical angles 60 60 5b – 5c = 60 5(13) -5c = 60 65 -5c = 60 -5c = -5 C = 1 60 b=13

Alternate Interior Angles = 25.) Solve for all the variables. Then find the measures of the angles. C=1 10c + d = 60 10(1) +d = 60 10 + d = 60 d = 50 60 60 60 60 Alternate Interior Angles =

25.) Solve for all the variables. Then find the measures of the angles. Linear pair Sum = 180 60 60 120 12d +6e +60 = 180 12d + 6e = 120 12(1) +6e=120 12 +6e = 120 6e = 108 e = 18 60 60 d = 1

25.) Solve for all the variables. Then find the measures of the angles. Alternate Interior Angles= 60 60 120 4f +4e = 120 4f +4(18) = 120 4f+72 = 120 4f = 48 f = 12 60 120 60 e = 18