3.2 Use Parallel Lines and Transversals

Objectives Use the properties of parallel lines to determine congruent angles Use algebra to find angle measures

Postulate 15 Corresponding  s Postulate If 2  lines are cut by a transversal, then each pair of corres.  s is . i.e. If l  m, then  1  2. l m 1 2

Theorem 3.1 Alternate Interior  s Theorem If 2  lines are cut by a transversal, then each pair of alternate interior  s is . i.e. If l  m, then  1  2. lmlm 1 2

Theorem 3.2 Alternate Exterior  s Theorem If 2  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

Theorem 3.3 Consecutive Interior  s Theorem If 2  lines are cut by a transversal, then each pair of consecutive int.  s is supplementary. i.e. If l  m, then  1 &  2 are supplementary or m  1 + m  2 = 180°. lmlm 1 2

If a transversal is  to one of 2  lines, then it is  to the other. i.e. If l  m, & t  l, then t  m. 1 2 Theorem 3.11  Transversal Theorem lmlm t

EXAMPLE 1 Identify congruent angles SOLUTION By the Corresponding Angles Postulate, m 5 = 120°. Using the Vertical Angles Congruence Theorem, m 4 = 120°. Because 4 and 8 are corresponding angles, by the Corresponding Angles Postulate, you know that m 8 = 120°. The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning.

EXAMPLE 2 Use properties of parallel lines ALGEBRA Find the value of x. SOLUTION By the Vertical Angles Congruence Theorem, m 4 = 115°. Lines a and b are parallel, so you can use the theorems about parallel lines. Consecutive Interior Angles Theorem m 4 + (x+5)° = 180° Substitute 115° for m ° + (x+5)° = 180° Combine like terms. x = 180 Subtract 120 from each side. x = 60

GUIDED PRACTICE for Examples 1 and 2 Vertical Angles Congruence Theorem. Corresponding Angles Postulate. m 5 =105° Alternate Exterior Angles Theorem m 8 =105° Use the diagram. 1. If m 1 = 105°, find m 4, m 5, and m 8. Tell which postulate or theorem you use in each case. m 4 = 105° ANSWER

GUIDED PRACTICE for Examples 1 and 2 Use the diagram. 2. If m 3 = 68° and m 8 = (2x + 4)°, what is the value of x ? Show your steps. m 3 =m x + 4 =180 2x + 72 =180 2x =108 x =54 m 7 + m 8 = 180 ANSWER

EXAMPLE 3 Prove the Alternate Interior Angles Theorem Prove that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. SOLUTION Draw a diagram. Label a pair of alternate interior angles as 1 and 2. You are looking for an angle that is related to both 1 and 2. Notice that one angle is a vertical angle with 2 and a corresponding angle with 1. Label it 3. GIVEN : p q PROVE :  1  2

EXAMPLE 3 Prove the Alternate Interior Angles Theorem STATEMENTS REASONS p q Given 2. 1  Corresponding Angles Postulate 3. 3  Vertical Angles Congruence Theorem 4. 1  2 Transitive Property of Congruence 4.4.

EXAMPLE 4 Solve a real-world problem Science When sunlight enters a drop of rain, different colors of light leave the drop at different angles. This process is what makes a rainbow. For violet light, m 2 = 40°. What is m 1 ? How do you know?

EXAMPLE 4 Solve a real-world problem SOLUTION Because the sun’s rays are parallel, 1 and 2 are alternate interior angles. By the Alternate Interior Angles Theorem, 1 2. By the definition of congruent angles, m 1 = m 2 = 40°.

GUIDED PRACTICE for Examples 3 and 4 3. In the proof in Example 3, if you use the third statement before the second statement, could you still prove the theorem? Explain. Yes; 3 and 2 congruence is not dependent on the congruence of 1 and 3. SAMPLE ANSWER

GUIDED PRACTICE for Examples 3 and 4 Suppose the diagram in Example 4 shows yellow light leaving a drop of rain. Yellow light leaves the drop at an angle of 41°. What is m 1 in this case? How do you know? 4. WHAT IF? 41°; 1 and 2 are alternate interior angles. By the Alternate Interior Angles Theorem, 1 2. By the definition of congruent angles, m 1 = m 2 = 41°. ANSWER

Assignment Pages 181– even; 24 – 29 all; 31-34,36,38,39 Draw diagrams for 38 and 39