Identify the type of angles. 1. 5, 7 ANSWER corresponding 2. 3, 6 ANSWER alternate interior 3. 1, 8 ANSWER alternate exterior

Use Properties of Parallel and Perpendicular Lines Target Use Properties of Parallel and Perpendicular Lines You will… Use angles formed by parallel lines and transversals.

VOCABULARY Corresponding Angles Postulate 15 – If two parallel lines are cut by a transversal, then pairs of corresponding angles are congruent. 1 Proof sequence: First, identify the angles as corresponding by the definition of corresponding angles; then state the angles are congruent by the Corresponding Angles Postulate 15.

VOCABULARY Alternate Interior Angles Theorem 3.1 – If two parallel lines are cut by a transversal, then pairs of alternate interior angles are congruent. Alternate Exterior Angles Theorem 3.2 – If two parallel lines are cut by a transversal, then pairs of alternate exterior angles are congruent. Consecutive Interior Angles Theorem 3.3 – If two parallel lines are cut by a transversal, then pairs of consecutive interior angles are supplementary. Proof sequence: First, identify the angle pair by type, then state the angles are congruent or supplementary by one of the theorems about parallel lines.

EXAMPLE 1 Identify congruent angles The measure of three of the numbered angles is 120°. Identify the angles. Explain your reasoning. 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°.

Use properties of parallel lines 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. m 4 + (x+5)° = 180° Consecutive Interior Angles Theorem 115° + (x+5)° = 180° Substitute 115° for m 4. x + 120 = 180 Combine like terms. x = 60 Subtract 120 from each side.

GUIDED PRACTICE for Examples 1 and 2 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 Vertical Angles Congruence Theorem. m 5 = 105° Corresponding Angles Postulate. m 8 = 105° Alternate Exterior Angles Theorem

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 7 + m 8 = 180 ANSWER m 3 = m 7 68 + 2x + 4 = 180 2x + 72 = 180 2x = 108 x = 54

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.

Prove the Alternate Interior Angles Theorem EXAMPLE 3 Prove the Alternate Interior Angles Theorem GIVEN : p q PROVE : ∠ 1 ∠ 2 STATEMENTS REASONS p q 1. 1. Given 2. Corresponding Angles Postulate 2. 1 ∠ 3 3. 3 ∠ 2 3. Vertical Angles Congruence Theorem 4. 1 ∠ 2 Transitive Property of Congruence 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°.