Presentation on theme: "Lesson 3.3, For use with pages"— Presentation transcript:
1Lesson 3.3, For use with pages 161-169 1. Find the value of x.ANSWER322. Write the converse of the following statement. If it is raining, then Josh needs an umbrella.ANSWERIf Josh needs an umbrella, then it is raining.
4Apply the Corresponding Angles Converse EXAMPLE 1Apply the Corresponding Angles ConverseALGEBRAFind the value of x that makes m n.SOLUTIONLines m and n are parallel if the marked corresponding angles are congruent.(3x + 5)o= 65oUse Postulate 16 to write an equation3x= 60Subtract 5 from each side.x= 20Divide each side by 3.The lines m and n are parallel when x = 20.
5GUIDED PRACTICEfor Example 1Is there enough information in the diagram to conclude that m n? Explain.ANSWERYes. m n because the angle corresponding to the angle measuring 75o also measures 75o since it forms a linear pair with the 105o angle. So, corresponding angles are congruent and Postulate 16 says the lines are parallel.
6GUIDED PRACTICEfor Example 1Explain why Postulate 16 is the converse of Postulate 15.ANSWERPostulate 16 switches the hypothesis and conclusion of Postulate 15.
7EXAMPLE 2Solve a real-world problemSnake PatternsHow can you tell whether the sides of the pattern are parallel in the photo of a diamond-back snake?SOLUTIONBecause the alternate interior angles are congruent, you know that the sides of the pattern are parallel.
8GUIDED PRACTICEfor Example 2Can you prove that lines a and b are parallel? Explain why or why not.Yes; Alternate Exterior Angles Converse.ANSWER
9GUIDED PRACTICEfor Example 2Can you prove that lines a and b are parallel? Explain why or why not.Yes; Corresponding Angles Converse.ANSWER
10GUIDED PRACTICEfor Example 2Can you prove that lines a and b are parallel? Explain why or why not.m m 2 = 180°No; Supplementary angles do not have to be congruent.ANSWER
11EXAMPLE 3Prove the Alternate Interior Angles ConverseProve that if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.SOLUTIONGIVEN :∠ ∠ 5PROVE :g h
12Prove the Alternate Interior Angles Converse EXAMPLE 3Prove the Alternate Interior Angles ConverseSTATEMENTSREASONS1.4 ∠1.Given2.1 ∠2.Vertical Angles Congruence Theorem3.1 ∠3.Transitive Property of Congruenceg h4.4.Corresponding Angles Converse
13EXAMPLE 4Write a paragraph proofIn the figure, r s and is congruent to Prove p q.SOLUTIONLook at the diagram to make a plan. The diagram suggests that you look at angles 1, 2, and 3. Also, you may find it helpful to focus on one pair of lines and one transversal at a time.
14EXAMPLE 4Write a paragraph proofPlan for Proofa.Look at 1 andLook at 2 andb.because r s.If then p q.
15EXAMPLE 4Write a paragraph proofPlan in Actiona.It is given that r s, so by the Corresponding Angles Postulate,b.It is also given that Then by the Transitive Property of Congruence for angles. Therefore, by the Alternate Interior Angles Converse, p q.
17EXAMPLE 5Use the Transitive Property of Parallel LinesThe flag of the United States has 13 alternating red and white stripes. Each stripe is parallel to the stripe immediately below it. Explain why the top stripe is parallel to the bottom stripe.U.S. Flag
18EXAMPLE 5Use the Transitive Property of Parallel LinesSOLUTIONThe stripes from top to bottom can be named s1, s2, s3, , s13. Each stripe is parallel to the one below it, so s1 s2, s2 s3, and so on. Then s1 s3 by the Transitive Property of Parallel Lines. Similarly, because s3 s4, it follows that s1 s4. By continuing this reasoning, s1 s13. So, the top stripe is parallel to the bottom stripe.
19GUIDED PRACTICEfor Examples 3, 4, and 5If you use the diagram at the right to prove the Alternate Exterior Angles Converse, what GIVEN and PROVE statements would you use?PROVE : j kGIVEN : ∠ANSWER
20GUIDED PRACTICEfor Examples 3, 4, and 5Copy and complete the following paragraph proof of the Alternate Interior Angles Converse using the diagram in Example 3.It is given that By the ? , Then by the Transitive Property of Congruence, ? . So, by the ? , g h.ANSWERIt is given that By the Vertical Angle Congruence, Then by the Transitive Property of Congruence, So, by the Corresponding Angle Converse Postulate , g h.
21GUIDED PRACTICEfor Examples 3, 4, and 5Each step is parallel to the step immediately above it. The bottom step is parallel to the ground. Explain why the top step is parallel to the ground.ANSWERAll of the steps are parallel. Since the bottom step is parallel to the ground, the Transitive Property of Parallel Lines applies, and the top step is parallel to the ground.