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Proving Lines Parallel: Angle Relationships and Theorems

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paloma-fields
25 diapositives · 428 Views · Nov 01, 2014

This lesson focuses on recognizing angle conditions that indicate parallel lines and proving that two lines are parallel based on specific angle relationships. Key concepts include the Converse of the Corresponding Angles Postulate, the Converse of the Alternate Exterior Angles Theorem, and the Converse of the Consecutive Interior Angles Theorem. Students will learn to apply these postulates and theorems through various proofs, examples, and algebraic methods to find missing angle measures important for determining line relationships.

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  1. 3.5 Proving Lines Parallel

  2. Objectives • Recognize angle conditions that occur with parallel lines • Prove that two lines are parallel based on given angle relationships

  3. Postulate 3.4Converse of the Corresponding Angles Postulate If two lines in a plane are cut by a transversal so that corresponding angles are congruent, then the lines are parallel. Abbreviation:If corr. s are , then lines are ║.

  4. Postulate 3.5Parallel Postulate If given a line and a point not on the line, then there exists exactly one line through the point that is parallel to the given line.

  5. Theorem 3.5Converse of the Alternate Exterior Angles Theorem If two lines in a plane are cut by a transversal so that a pair of alternate exterior angles is congruent, then the two lines are parallel. Abbreviation:If alt ext. s are , then lines are ║.

  6. Theorem 3.6Converse of the Consecutive Interior Angles Theorem If two lines in a plane are cut by a transversal so that a pair of consecutive interior angles is supplementary, then the lines are parallel. Abbreviation:If cons. int. s are supp., then lines are ║.

  7. Proof of the Converse of the Consecutive Interior Angles Theorem Given:4 and 5 are supplementary Prove: g ║ h g 6 5 h 4

  8. Paragraph Proof of the Converse of the Consecutive Interior Angles Theorem You are given that 4 and 5 are supplementary. By the Supplement Theorem, 5 and 6 are also supplementary because they form a linear pair. If 2 s are supplementary to the same , then 4  6. Therefore, by the Converse of the Corresponding s Angles Postulate, g and h are parallel.

  9. Theorem 3.7Converse of the Alternate Interior Angles Theorem If two lines in a plane are cut by a transversal so that a pair of alternate interior angles is congruent, then the lines are parallel. Abbreviation:If alt. int. s are , then lines are ║.

  10. Proof of the Converse of the Alternate Interior Angles Theorem Given: 1  2 Prove: m ║ n m 3 2 1 n

  11. Statements: 1  2 2  3 1  3 m ║ n Reasons: Given Vertical Angles are  Transitive prop. If corres. s are , then lines are ║ Two - Column Proof of the Converse of the Alternate Interior Angles Theorem

  12. Theorem 3.8 In a plane, if two lines are perpendicular to the same line, then they are parallel. Abbreviation:If 2 lines are ┴ to the same line, then the lines are ║.

  13. consecutive interior angles are supplementary. So, consecutive interior angles are not supplementary. So, c is not parallel to a or b. Example 1: Determine which lines, if any, are parallel. Answer:

  14. Your Turn: Determine which lines, if any, are parallel. Answer:

  15. ALGEBRA Find x and mZYN so that ExploreFrom the figure, you know that and You also know that are alternate exterior angles. Example 2:

  16. PlanFor line PQ to be parallel to MN, the alternate exterior angles must be congruent. Substitute the given angle measures into this equation and solve for x. Once you know the value of x, use substitution to find Example 2: Solve Alternate exterior angles Substitution Subtract 7x from each side. Add 25 to each side. Divide each side by 4.

  17. ExamineVerify the angle measure by using the value of x to find Since Answer: Example 2: Original equation Simplify.

  18. ALGEBRAFind x and mGBA so that Answer: Your Turn:

  19. Given: Prove: Example 3:

  20. Statements Reasons 1. 1.Given 2. 2. Consecutive InteriorThm. 3. Def. of congruents 3. 7. 7. If cons. int.s are suppl.,thenlines are . Example 3: Proof: ; 4. 7 + 6 = 1804.Def. Suppl. s 5. 4 + 6 = 1805.Substitution 6. 4 and 6 are suppl6. Def. Suppl. s

  21. Your Turn: Given: a || b 1 12 Prove:x || y

  22. Your Turn: Proof: Statements Reasons 1.a || b; 1 12 1. Given 2.1  13 2. Corres. s Postulate 3. 13  15 3. Vertical s are  4. 12  15 4. Substitution 5. x || y 5. If corres. s are , then lines are ||

  23. Answer: Example 4:

  24. Your Turn: Answer: Since the slopes are not equal, r is not parallel to s.

  25. Assignment • Geometry: Pg. 155 #13 – 31, 34, 35 • Pre-AP Geometry: Pg. 155 #13 – 31, 34 - 39