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Parallel Lines and Transversals Lesson 3.4 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. Review: Parallel Lines Transversal • 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) r w r || w • 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 . 1 2 l • i.e. If l m, then 12. m Corresponding Angles • Look for angles in an F shape to help you find corresponding angles. • If two parallel lines are cut by a transversal, then Corresponding angles are congruent. 1 r 2 3 4 5 6 w 7 r || w 8 Example 1 Find the measure of the numbered angle. a. b. c. b. m5 = 135° c. m2 = 90° SOLUTION a. m6 = 60° Your Turn: Find the measure of the numbered angle. 1. ANSWER 120° ANSWER 145° ANSWER 45° 2. 3. 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 Answer: and Find Theorem 3.5 Alternate Interior ’s Theorem • If 2 parallel lines are cut by a transversal, then each pair of alternate interior ’s is . l 1 2 m • i.e. If l m, then 12. Alternate Interior Angles • Look for angles inside a Z or N shape to find alternate interior angles. • If two parallel lines are cut by a transversal, then Alternate Interior angles are congruent. r 3 4 5 w r || w 6 Example 3 Find the measure of PQR. a. b. c. b. mPQR = 120° c. mPQR = 70° SOLUTION a. mPQR = 35° Your Turn: Find the measure of the numbered angle. 1. ANSWER 90° ANSWER 65° ANSWER 100° 2. 3. Theorem 3.6 Alternate Exterior ’s Theorem • If 2 parallel lines are cut by a transversal, then the pairs of alternate exterior ’s are . l m 1 2 • i.e. If l m, then 12. • If two parallel lines are cut by a transversal, then Alternate Exterior Angles are congruent. 1 2 r w 7 r || w 8 Example 4 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. m1 + m2 = 180° m1 + 75° = 180° m1 + 75° – 75° = 180° – 75° m1 = 105° Linear Pair Postulate Substitute 75° for m2. Subtract 75° from each side. Simplify. Your Turn: Find the measure of the numbered angle. 1. ANSWER 130° ANSWER 42° ANSWER 90° 2. 3. Your Turn: Use the diagram. Tell whether the angles are congruent or not congruent. Explain. 4. 1 and 8 ANSWER congruent by the Alternate Exterior Angles Theorem 5. 3 and 4 ANSWER Not congruent; the angles are a linear pair. 6. 4 and 2 ANSWER Not congruent; the angles are a linear pair. Your Turn: Use the diagram. Tell whether the angles are congruent or not congruent. Explain. 7. 2 and 7 ANSWER congruent by the Alternate Exterior Angles Theorem 8. 3 and 7 ANSWER congruent by the Corresponding Angles Postulate 9. 3 and 8 ANSWER Not congruent; there is no special relationship between these angles. 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. l 1 m 2 • i.e. If l m, then 1 & 2 are supplementary or m1 + m2 = 180°. Same-Side Interior Angles • Look for angles inside a C shape to find same-side interior angles. • If two parallel lines are cut by a transversal, then each pair of Same-Side Interior Angles is supplementary. r 3 4 5 w r || w 6 Example 5 Find the measure of the numbered angle. a. b. SOLUTION a. m5 + 80° = 180° m5 = 100° b. m6 + 130° = 180° m6 = 50° Example 6 Find the value of x. SOLUTION (x + 15)° = 125° x = 110 Corresponding Angles Postulate Subtract 15 from each side. Your Turn: Find the value of x. 1. ANSWER 85 ANSWER 104 ANSWER 40 2. 3. • 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. 11 1 lr 2 3 4 4 5 5 lw 6 7 88 58 58 ˚˚ r 2 3 458 ˚ 58˚ 585 ˚ w 6 7 8 58˚ r || w 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 58 ˚ r 122˚ 2 3122 ˚ 4 5 w 6 ˚ 122 7 8 122 ˚ r || w 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: ALGEBRA If 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: and ALGEBRA If find x and y. Answer: Example 9: 1 125o Find: m1 = m2 = m3 = m4 = m5 = m6 = x= 55° 125° 55° 2 3 5 4 125° 55° 125° 40° x+15o 6 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