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OTCQ Given:  ABC is a straight angle and BD ABC. Prove  ABD is a right angle.

OTCQ Given:  ABC is a straight angle and BD ABC. Prove  ABD is a right angle. D. Statements. Reasons. A. B. C. OTCQ # 100609 Given:  ABC is a straight angle and BD ABC. Prove  ABD is a right angle. D. Statements. Reasons. A. B. C. 1.Given.

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OTCQ Given:  ABC is a straight angle and BD ABC. Prove  ABD is a right angle.

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  1. OTCQ Given:  ABC is a straight angle and BD ABC. Prove  ABD is a right angle. D Statements Reasons A B C

  2. OTCQ # 100609 Given:  ABC is a straight angle and BD ABC. Prove  ABD is a right angle. D Statements Reasons A B C 1.Given. 1.  ABC is a straight angle and BD ABC. Conclusion:  ABD is a right angle

  3. OTCQ # 100609 Given:  ABC is a straight angle and BD ABC. Prove  ABD is a right angle. D Statements Reasons A B C 1.Given. 2. Definition of perpendicular. QED. Quo era demonstratum 1.  ABC is a straight angle and BD ABC. 2. If BD ABC, then the intersection of BD and ABC forms 2 right angles:  ABD and  DBC Conclusion:  ABD is a right angle

  4. Aim 4-3 How do we prove theorems about angles (part 2)? GG 30, GG 32, GG 34

  5. OBJECTIVE SWBAT prove angle theorems. SWBAT recall some basic definitions.

  6. Prove Theorem 4-2 If two angles are straight angles, then they are congruent. A B C Given ABC is a straight angle and DEF is a Straight angle. Prove ABC  DEF. D E F Statements Reasons • Given • Definition of straight • angle. • 3.Definition of straight • angle. • 4.Definition of • congruent. QED • ABC is a straight angle and DEF is a straight angle. • m ABC = 180○ • m DEF = 180○. • Conclusion ABC  DEF.

  7. Theorem 4-1: If 2 angles are right angles, then they are congruent. Theorem 4-1: If 2 angles are straight angles, then they are congruent. Adjacent angles: are two angles in the same plane that have a common vertex and a common side, but do not have any interior points in common.

  8. Complementary angles are two angles the sum of whose degree measures is 90○. Supplementary angles are two angles the sum of whose degree measures is 180○.

  9. Theorem: If 2 angles are complements of the same angle then they are congruent. Why?

  10. Theorem: If 2 angles are complements of the same angle then they are congruent. Why? Given m 1= 45○ m 2= 45○ m 3= 45○ 3 1 2

  11. Theorem: If 2 angles are complements of the same angle then they are congruent. Why? Given m 1= 45○ m 2= 45○ m 3= 45○ 3 1 2 m1+ m 2= 90○ , hence  2 is the complement of  1. m1+ m 3= 90○ , hence  3 is the complement of  1. Since 2 and 3 are each the complement of  1, then 2 and 3 must be congruent.

  12. Theorem: If 2 angles are congruent then their complements are congruent. Why?

  13. Theorem: If 2 angles are congruent then their complements are congruent. Why? Given m 1= 30○ m 2= 30○ 3 4 1 2

  14. Theorem: If 2 angles are congruent then their complements are congruent. Why? Given m 1= 30○ m 2= 30○ If  3 is complementary to 1, what is the degree measure of 3? If  4 is complementary to 2, what is the degree measure of 4? 4 3 2 1

  15. Theorem: If 2 angles are congruent then their complements are congruent. Why? Given m 1= 30○ m 2= 30○ If  3 is complementary to 1, what is the degree measure of 3? (90○ - 30○ = 60○) If  4 is complementary to 2, what is the degree measure of 4? 4 3 2 1

  16. Theorem 4-4: If 2 angles are congruent then their complements are congruent. Why?  3 4 Given m 1= 30○ m 2= 30○ If  3 is complementary to 1, what is the degree measure of 3? (90○ - 30○ = 60○) If  4 is complementary to 2, what is the degree measure of 4? (90○ - 30○ = 60○) 4 3 2 1

  17. Theorem: If 2 angles are supplements of the same angle then they are congruent. Why? Please try to draw 2 angles that are supplementary to the same angle.

  18. E Theorem: If 2 angles are supplements of the same angle then they are congruent. Given: ABC is a straight angle, we can say that ABE is a supplement to EBC. A B D C

  19. E Theorem: If 2 angles are supplements of the same angle then they are congruent. Given: ABC is a straight angle, we can say that ABE is a supplement to EBC. A B D C Next, given that DBE is a straight angle, we can say that DBC is a supplement to EBC.

  20. Conclusion: ABE  DBC E Theorem: If 2 angles are supplements of the same angle then they are congruent. Given: ABC is a straight angle, we can say that ABE is a supplement to EBC. A B D C Next, given that DBE is a straight angle, we can say that DBC is a supplement to EBC.

  21. Conclusion: ABE  DBC E Theorem: If 2 angles are supplements of the same angle then they are congruent. Given: ABC is a straight angle, we can say that ABE is a supplement to EBC. A 65○ 115○ B 65○ D C Next, given that DBE is a straight angle, we can say that DBC is a supplement to EBC.

  22. Theorem: If 2 angles are congruent then their supplements are congruent. Why?

  23. Conclusion: ABD  EBC E Theorem: If 2 angles are congruent then their supplements are congruent. Given: ABC is a straight angle. DBE is a straight angle. ABE  DBC A 65○ 115○ B 65○ D C

  24. Conclusion: ABD  EBC E Theorem: If 2 angles are congruent then their supplements are congruent. Given: ABC is a straight angle. DBE is a straight angle. ABE  DBC A 65○ 115○ 115○ B 65○ D C

  25. E Linear pair of angles: 2 adjacent angles whose sum is a straight angle. ABE and EBC are a linear pair of angles. The others? A 65○ 115○ 115○ B 65○ D C

  26. Why 4 pairs of linear pairs? E Linear pair of angles: 2 adjacent angles whose sum is a straight angle. ABE and EBC are a linear pair of angles. The others? EBC and CBD. CBD and DBA. DBA and ABE. There should always be 4 pairs of linear pairs when 2 lines intersect. A 65○ 115○ 115○ B 65○ D C

  27. Theorem: Linear pairs of angles are supplementary. E Linear pair of angles: 2 adjacent angles whose sum is a straight angle. ABE and EBC are a linear pair of angles. The others? EBC and CBD. CBD and DBA. DBA and ABE. There should always be 4 pairs of linear pairs when 2 lines intersect. A 65○ 115○ 115○ B 65○ D C

  28. Theorem: If 2 lines intersect to form congruent adjacent angles, then they are perpendicular. 1 2 4 3

  29. Theorem: If 2 lines intersect to form congruent adjacent angles, then they are perpendicular. Since m1 + m 2 =180○ and  1 2, we may substitute to say m 1 + m 1 =180○ and then 2 m 1=180○ and then 2 m 1=180○ and then 2 2 m 1=90○ We can do the same for  2,  3 and  4 1 2 4 3

  30. E Vertical angles: 2 angles in which the sides of one angle are opposite rays to the sides of the second angle. Theorem 4-9. If two lines intersect, then the vertical angles are congruent. Vertical angles: EBC and ABD. ABE and DBC. There should always be 2 pairs of vertical angles pairs when 2 lines intersect. A 65○ 115○ 115○ B 65○ D C

  31. B F G A 4 1 C E 2 3 Statements Reasons D H Given: ADC   EHG 1  4 Prove: 2  3

  32. Recall Properties of Equality 1) Reflexive: a = a 2) Symmetric: If a = b then b = a. • Transitive: If a = b and b = c, then a = c. • Substitution: If a = b, then a can be replaced by b.

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