Parallel Lines and Transversals Lesson 3.4 w r 8 2 4 3 5 6 7 1 r || w

Objective: Find the congruent angles formed when a transversal cuts parallel lines.

Key Vocabulary None

Postulates 8 - Corresponding Angles

Theorems 3.5 Alternate Interior Angles 3.6 Alternate Exterior Angles 3.7 Same-Side Interior Angles

Parallel Lines and Angle Pairs Line 𝓏 is a transversal of parallel lines 𝓍 and 𝓎. Since lines 𝓍 and 𝓎 are parallel, there are special relationships between specific pairs of angles.

Parallel Lines Transversal Review: Parallel Lines Transversal

PARALLEL LINES Two or more lines are parallel if and only if they are in the same plane and they do not intersect. (line w and r) w r r || w

TRANSVERSAL A line intersecting two or more coplanar lines. (lines r and w) r w r || w

Postulate 8 Corresponding ’s Postulate If 2 parallel lines are cut by a transversal, then each pair of corresponding ’s is . i.e. If l m, then 12. 1 2 l m

Corresponding Angles Look for angles in an F shape to help you find corresponding angles.

CORRESPONDING ANGLES If two parallel lines are cut by a transversal, then Corresponding angles are congruent. r w r || w 1 5 2 6 3 7 4 8

Example 1 Find the measure of the numbered angle. SOLUTION a. c. SOLUTION a. m6 = 60° b. m5 = 135° c. m2 = 90° 13

Your Turn: Find the measure of the numbered angle. ANSWER 120° ANSWER 1. ANSWER 120° 2. ANSWER 145° 3. ANSWER 45°

Example 2: In the figure and Find Corresponding Angles Postulate Vertical Angles Postulate Transitive Property Definition of congruent angles Substitution Answer:

Your Turn: In the figure and Find Answer:

Theorem 3.5 Alternate Interior ’s Theorem If 2 parallel lines are cut by a transversal, then each pair of alternate interior ’s is . i.e. If l m, then 12. 1 2 l m

Alternate Interior Angles Look for angles inside a Z or N shape to find alternate interior angles.

ALTERNATE INTERIOR ANGELS If two parallel lines are cut by a transversal, then Alternate Interior angles are congruent. w r r || w 3 4 5 6

Example 3 Find the measure of PQR. SOLUTION a. mPQR = 35° b. c. SOLUTION a. mPQR = 35° b. mPQR = 120° c. mPQR = 70° 20

Your Turn: Find the measure of the numbered angle. ANSWER 90° ANSWER 1. ANSWER 90° 2. ANSWER 65° 3. ANSWER 100°

Theorem 3.6 Alternate Exterior ’s Theorem If 2 parallel lines are cut by a transversal, then the pairs of alternate exterior ’s are . i.e. If l m, then 12. l m 1 2

ALTERNATE EXTERIOR ANGLES If two parallel lines are cut by a transversal, then Alternate Exterior Angles are congruent. w r r || w 1 2 7 8

Example 4 Linear Pair Postulate Substitute 75° for m2. Find the measures of 1 and 2. SOLUTION The measure of 2 is 75° because alternate exterior angles are congruent. The measure of 2 can be used to find the measure of 1. m1 + m2 = 180° Linear Pair Postulate m1 + 75° = 180° Substitute 75° for m2. m1 + 75° – 75° = 180° – 75° Subtract 75° from each side. m1 = 105° Simplify. 24

Your Turn: Find the measure of the numbered angle. ANSWER 130° ANSWER 1. ANSWER 130° 2. ANSWER 42° 3. ANSWER 90°

Your Turn: Use the diagram. Tell whether the angles are congruent or not congruent. Explain. ANSWER congruent by the Alternate Exterior Angles Theorem 4. 1 and 8 ANSWER Not congruent; the angles are a linear pair. 5. 3 and 4 ANSWER Not congruent; the angles are a linear pair. 6. 4 and 2

Your Turn: Use the diagram. Tell whether the angles are congruent or not congruent. Explain. ANSWER congruent by the Alternate Exterior Angles Theorem 7. 2 and 7 ANSWER congruent by the Corresponding Angles Postulate 8. 3 and 7 ANSWER Not congruent; there is no special relationship between these angles. 9. 3 and 8

Theorem 3.7 Same-Side Interior ’s Theorem If 2 parallel lines are cut by a transversal, then each pair of same-side interior ’s is supplementary. i.e. If l m, then 1 & 2 are supplementary or m1 + m2 = 180°. l m 1 2

Same-Side Interior Angles Look for angles inside a C shape to find same-side interior angles.

Same-side INTERIOR ANGELS If two parallel lines are cut by a transversal, then each pair of Same-Side Interior Angles is supplementary. w r r || w 3 4 5 6

Example 5 Find the measure of the numbered angle. SOLUTION a. 31

Example 6 Find the value of x. Corresponding Angles Postulate SOLUTION (x + 15)° = 125° Corresponding Angles Postulate x = 110 Subtract 15 from each side. 32

Your Turn: Find the value of x. 1. ANSWER 85 2. ANSWER 104 3. ANSWER 40

How to Find Angle Measurements On Two Parallel Lines Cut By A Transversal w r 8 2 4 3 5 6 7 1

PARALLEL LINES If two parallel lines are cut by a transversal, then …… Corresponding angles are congruent, Alternate Interior angles are congruent, . . . . And . . . . Alternate Exterior angles are congruent. lw lr 4 1 3 2 5 6 7 8 1 5 1 4 5 8

Corresponding and Vertical Angles r w r || w 2 3 4 6 5 7 8 58 ˚ 58 ˚ 58 ˚ 58˚ 58 ˚ 58˚ If angle 1 = 58 ˚ then angle 5 = 58 ˚ because they are corresponding angles, which are congruent to each other Since angle 1 = 58 ˚ then angle 4 = 58 ˚ and since angle 5 = 58 ˚ then angle 8 = 58 ˚ because they are vertical angles, which are congruent to each other

Alternate Exterior and Interior Angles r w r || w 2 3 4 6 5 7 8 58 ˚ 122˚ 122 ˚ 122 ˚ 122˚ If angle 1 = 58 ˚ then angle 2 = 122 ˚ because the two angles form a line, which is equal to 180 ˚ Since angle 2 = 122 ˚ then angle 7 = 122 ˚ because they are Alternate Exterior angles, which are congruent to each other. Since angle 3 = 122 ˚ then angle 6 = 122 ˚ because they are Alternate Interior angles, which are congruent to each other.

Example 7: What is the measure of RTV?

Example 7: Alternate Interior Angles Theorem Definition of congruent angles Substitution

Example 7: Alternate Interior Angles Theorem Definition of congruent angles Substitution Angle Addition Postulate Answer: RTV = 125°

Example 8: If ALGEBRA and find x and y. Find x. by the Corresponding Angles Postulate.

Example 8: Definition of congruent angles Substitution Subtract x from each side and add 10 to each side. Find y. by the Alternate Exterior Angles Theorem. Definition of congruent angles Substitution

Example 8: Simplify. Add 100 to each side. Divide each side by 4. Answer:

Your Turn: ALGEBRA If and find x and y. Answer:

Example 9: 55° 125° 40° m1 = m2 = m3 = m4 = m5 = m6 = x = Find: 125o 2 Find: m1 = m2 = m3 = m4 = m5 = m6 = x = 55° 125° 40° 3 5 4 6 x+15o

Joke Time What flower grows between your nose and your chin? Tulips How many sides are there to a circle? 2 – inside and outside. What do you get when you cross an elephant and Darth Vader? An elevader.

Assignment Section 3.4, pg. 132-135: #1-12 all, 15-55 odd