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OTCQ What is the measure of one angle in a equilateral/equiangular triangle?

OTCQ What is the measure of one angle in a equilateral/equiangular triangle?. Aim 4-3 How do we prove theorems about angles (part 1)? GG 30, GG 32, GG 34. Objectives SWBAT to construct the form of a proof and SWBAT to conjecture about appropriate statements. Starters:

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OTCQ What is the measure of one angle in a equilateral/equiangular triangle?

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  1. OTCQWhat is the measure of one angle in a equilateral/equiangulartriangle?

  2. Aim 4-3 How do we prove theorems about angles (part 1)? GG 30, GG 32, GG 34

  3. Objectives SWBAT to construct the form of a proof and SWBAT to conjecture about appropriate statements.

  4. Starters: 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. How could we justify these statements in a proof?

  5. Prove Theorem 4-2 If two angles are straight angles, then they are congruent. Given ABC is a straight angle and DEF is a straight angle. Prove ABC  DEF. A B C D E F Statements Reasons

  6. 4-3 #17.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. 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○.

  8. Theorem 4-3: If 2 angles are complements of the same angle then they are congruent. Why?

  9. Theorem 4-3: 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

  10. Theorem 4-3: 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.

  11. Theorem 4-4: If 2 angles are congruent then their complements are congruent. Why?

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

  13. Theorem 4-4: 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

  14. Theorem 4-4: 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

  15. 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

  16. Theorem 4-5: 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.

  17. E Theorem 4-5: 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

  18. E Theorem 4-5: 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.

  19. Conclusion: ABE  DBC E Theorem 4-5: 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 4-5: 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.

  21. Theorem 4-6: If 2 angles are congruent then their supplements are congruent. Why?

  22. Conclusion: ABD  EBC E Theorem 4-6: 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

  23. Conclusion: ABD  EBC E Theorem 4-6: 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

  24. 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

  25. 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

  26. Theorem 4-7: 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

  27. Theorem 4-8: If 2 lines intersect to form congruent adjacent angles, then they are perpendicular. 1 2 4 3

  28. Theorem 4-8: 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

  29. 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

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