Geometry Section 2-8 1112

4,107 views

Published on

Published in: Education, Technology, Spiritual
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
4,107
On SlideShare
0
From Embeds
0
Number of Embeds
3,042
Actions
Shares
0
Downloads
10
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

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

×