Presentation on theme: "EXAMPLE 3 Prove the Alternate Interior Angles Theorem"— Presentation transcript:
1EXAMPLE 3Prove the Alternate Interior Angles TheoremProve that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.SOLUTIONDraw a diagram. Label a pair of alternate interior angles as 1 and You are looking for an angle that is related to both 1 and Notice that one angle is a vertical angle with and a corresponding angle with Label itGIVEN :p qPROVE :∠ ∠
2Prove the Alternate Interior Angles Theorem EXAMPLE 3Prove the Alternate Interior Angles TheoremSTATEMENTSREASONSp q1.1.Given2.Corresponding Angles Postulate2.1 ∠3.3 ∠3.Vertical Angles Congruence Theorem4.1 ∠Transitive Property of Congruence4.
3EXAMPLE 4Solve a real-world problemScienceWhen 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 = 40°. What is m 1? How do you know?
4EXAMPLE 4Solve a real-world problemSOLUTIONBecause the sun’s rays are parallel, 1 and 2 are alternate interior angles. By the Alternate Interior Angles Theorem, By the definition of congruent angles, m = m = 40°.
5GUIDED PRACTICEfor Examples 3 and 43.In the proof in Example 3, if you use the third statement before the second statement, could you still prove the theorem? Explain.SOLUTIONYes still we can prove the theorem.As and congruence is not dependent on the congruence of and
6GUIDED PRACTICEfor Examples 3 and 44.WHAT IF?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 in this case? How do you know?SOLUTIONBecause the sun’s rays are parallel, 1 and 2 are alternate interior angles. By the Alternate Interior Angles Theorem, By the definition of congruent angles, m = m = 41°.