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# Geometry Section 2-8 1112

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### Geometry Section 2-8 1112

1. 1. Section 2-8 Proving Angle RelationshipsWednesday, November 16, 2011
2. 2. Essential Questions How do you write proofs involving supplementary and complementary angles? How do you write proofs involving congruent and right angles?Wednesday, November 16, 2011
3. 3. More Postulates and Theorems Protractor Postulate: Angle Addition Postulate: Theorem 2.3 - Supplement Theorem:Wednesday, November 16, 2011
4. 4. More Postulates and Theorems Protractor Postulate: Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180 Angle Addition Postulate: Theorem 2.3 - Supplement Theorem:Wednesday, November 16, 2011
5. 5. More Postulates and Theorems Protractor Postulate: Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180 This means we can measure angles in degrees Angle Addition Postulate: Theorem 2.3 - Supplement Theorem:Wednesday, November 16, 2011
6. 6. More Postulates and Theorems Protractor Postulate: Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180 This means we can measure angles in degrees Angle Addition Postulate: D is in the interior of ∠ABC IFF m∠ABD + m∠DBC = m∠ABC Theorem 2.3 - Supplement Theorem:Wednesday, November 16, 2011
7. 7. More Postulates and Theorems Protractor Postulate: Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180 This means we can measure angles in degrees Angle Addition Postulate: D is in the interior of ∠ABC IFF m∠ABD + m∠DBC = m∠ABC Theorem 2.3 - Supplement Theorem: If two angles form a linear pair, then they are supplementary anglesWednesday, November 16, 2011
8. 8. More Postulates and Theorems Theorem 2.4 - Complement Theorem: Theorem 2.5 - Properties of Angle Congruence Reﬂexive Property of Congruence: Symmetric Property of Congruence: Transitive Property of Congruence:Wednesday, November 16, 2011
9. 9. More Postulates and Theorems Theorem 2.4 - Complement Theorem: If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles Theorem 2.5 - Properties of Angle Congruence Reﬂexive Property of Congruence: Symmetric Property of Congruence: Transitive Property of Congruence:Wednesday, November 16, 2011
10. 10. More Postulates and Theorems Theorem 2.4 - Complement Theorem: If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles Theorem 2.5 - Properties of Angle Congruence Reﬂexive Property of Congruence: ∠1 ≅ ∠1 Symmetric Property of Congruence: Transitive Property of Congruence:Wednesday, November 16, 2011
11. 11. More Postulates and Theorems Theorem 2.4 - Complement Theorem: If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles Theorem 2.5 - Properties of Angle Congruence Reﬂexive Property of Congruence: ∠1 ≅ ∠1 Symmetric Property of Congruence: If ∠1 ≅ ∠2, then ∠2 ≅ ∠1 Transitive Property of Congruence:Wednesday, November 16, 2011
12. 12. More Postulates and Theorems Theorem 2.4 - Complement Theorem: If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles Theorem 2.5 - Properties of Angle Congruence Reﬂexive Property of Congruence: ∠1 ≅ ∠1 Symmetric Property of Congruence: If ∠1 ≅ ∠2, then ∠2 ≅ ∠1 Transitive Property of Congruence: If ∠1 ≅ ∠2 and ∠2 ≅ ∠3, then ∠1 ≅ ∠3Wednesday, November 16, 2011
13. 13. More Postulates and Theorems Theorem 2.6 - Congruent Supplements Theorem: Theorem 2.7 - Congruent Complements Theorem: Theorem 2.8 - Vertical Angles Theorem:Wednesday, November 16, 2011
14. 14. More Postulates and Theorems Theorem 2.6 - Congruent Supplements Theorem: Angles supplementary to the same angle or to congruent angles are congruent Theorem 2.7 - Congruent Complements Theorem: Theorem 2.8 - Vertical Angles Theorem:Wednesday, November 16, 2011
15. 15. More Postulates and Theorems Theorem 2.6 - Congruent Supplements Theorem: Angles supplementary to the same angle or to congruent angles are congruent Theorem 2.7 - Congruent Complements Theorem: Angles complementary to the same angle or to congruent angles are congruent Theorem 2.8 - Vertical Angles Theorem:Wednesday, November 16, 2011
16. 16. More Postulates and Theorems Theorem 2.6 - Congruent Supplements Theorem: Angles supplementary to the same angle or to congruent angles are congruent Theorem 2.7 - Congruent Complements Theorem: Angles complementary to the same angle or to congruent angles are congruent Theorem 2.8 - Vertical Angles Theorem: If two angles are vertical angles, then they are congruentWednesday, November 16, 2011
17. 17. EN EV More Postulates and Theorems Right Angle Theorems Theorem 2.9: Theorem 2.10: Theorem 2.11: Theorem 2.12: Theorem 2.13:Wednesday, November 16, 2011
18. 18. EN EV More Postulates and Theorems Right Angle Theorems Theorem 2.9: Perpendicular lines intersect to form four right angles Theorem 2.10: Theorem 2.11: Theorem 2.12: Theorem 2.13:Wednesday, November 16, 2011
19. 19. EN EV More Postulates and Theorems Right Angle Theorems Theorem 2.9: Perpendicular lines intersect to form four right angles Theorem 2.10: All right angles are congruent Theorem 2.11: Theorem 2.12: Theorem 2.13:Wednesday, November 16, 2011
20. 20. EN EV More Postulates and Theorems Right Angle Theorems Theorem 2.9: Perpendicular lines intersect to form four right angles Theorem 2.10: All right angles are congruent Theorem 2.11: Perpendicular lines form congruent adjacent angles Theorem 2.12: Theorem 2.13:Wednesday, November 16, 2011
21. 21. EN EV More Postulates and Theorems Right Angle Theorems Theorem 2.9: Perpendicular lines intersect to form four right angles Theorem 2.10: All right angles are congruent Theorem 2.11: Perpendicular lines form congruent adjacent angles Theorem 2.12: If two angles are congruent and supplementary, then each angle is a right angle Theorem 2.13:Wednesday, November 16, 2011
22. 22. EN EV More Postulates and Theorems Right Angle Theorems Theorem 2.9: Perpendicular lines intersect to form four right angles Theorem 2.10: All right angles are congruent Theorem 2.11: Perpendicular lines form congruent adjacent angles Theorem 2.12: If two angles are congruent and supplementary, then each angle is a right angle Theorem 2.13: If two congruent angles form a linear pair, then they are right anglesWednesday, November 16, 2011
23. 23. Example 1 Using a protractor, a construction worker measures that the angle a beam makes with the ceiling is 42°. What is the measure of the angle that the beam makes with the wa%?Wednesday, November 16, 2011
24. 24. Example 1 Using a protractor, a construction worker measures that the angle a beam makes with the ceiling is 42°. What is the measure of the angle that the beam makes with the Ceiling wa%? 42° Be am Wa!Wednesday, November 16, 2011
25. 25. Example 1 Using a protractor, a construction worker measures that the angle a beam makes with the ceiling is 42°. What is the measure of the angle that the beam makes with the Ceiling wa%? 42° Be am 90°-42° Wa!Wednesday, November 16, 2011
26. 26. Example 1 Using a protractor, a construction worker measures that the angle a beam makes with the ceiling is 42°. What is the measure of the angle that the beam makes with the Ceiling wa%? 42° Be am 90°-42° Wa! 48°Wednesday, November 16, 2011
27. 27. Example 2 At 4:00 on an analog clock, the angle between the hour and minute hands of a clock is 120°. When the second hand bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and the hour hands?Wednesday, November 16, 2011
28. 28. Example 2 At 4:00 on an analog clock, the angle between the hour and minute hands of a clock is 120°. When the second hand bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and the hour hands? Since the larger angle of 120° is bisected, two smaller angles of 60° are formed, and since those two angles add up to the larger one, both angles we are looking for are 60°.Wednesday, November 16, 2011
29. 29. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent.Wednesday, November 16, 2011
30. 30. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent.Wednesday, November 16, 2011
31. 31. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent. ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°Wednesday, November 16, 2011
32. 32. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent. ∠1 and ∠4 form a linear pair, Given and m∠3 + m∠1 = 180°Wednesday, November 16, 2011
33. 33. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent. ∠1 and ∠4 form a linear pair, Given and m∠3 + m∠1 = 180° ∠1 and ∠4 are supplementaryWednesday, November 16, 2011
34. 34. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent. ∠1 and ∠4 form a linear pair, Given and m∠3 + m∠1 = 180° Linear pairs are ∠1 and ∠4 are supplementary supplementaryWednesday, November 16, 2011
35. 35. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent. ∠1 and ∠4 form a linear pair, Given and m∠3 + m∠1 = 180° Linear pairs are ∠1 and ∠4 are supplementary supplementary ∠3 and ∠1 are supplementaryWednesday, November 16, 2011
36. 36. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent. ∠1 and ∠4 form a linear pair, Given and m∠3 + m∠1 = 180° Linear pairs are ∠1 and ∠4 are supplementary supplementary Def. of ∠3 and ∠1 are supplementary supplementaryWednesday, November 16, 2011
37. 37. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent. ∠1 and ∠4 form a linear pair, Given and m∠3 + m∠1 = 180° Linear pairs are ∠1 and ∠4 are supplementary supplementary Def. of ∠3 and ∠1 are supplementary supplementary ∠3 ≅ ∠4Wednesday, November 16, 2011
38. 38. Example 3 In the ﬁgure, ∠1 and ∠4 form a linear pair, and m∠3 + m∠1 = 180°. Prove that ∠3 and ∠4 are congruent. ∠1 and ∠4 form a linear pair, Given and m∠3 + m∠1 = 180° Linear pairs are ∠1 and ∠4 are supplementary supplementary Def. of ∠3 and ∠1 are supplementary supplementary ∠3 ≅ ∠4 Angles supplementary to same ∠ are ≅Wednesday, November 16, 2011
39. 39. Check Your Understanding Check out problems #1-7 on page 154 to see what you understand (or don’t) and formulate some questions on the ideas.Wednesday, November 16, 2011
40. 40. Problem SetWednesday, November 16, 2011
41. 41. Problem Set p. 154 #8-20 “Compassion for others begins with kindness to ourselves.” - Pema ChodronWednesday, November 16, 2011