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: 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?

Prove Theorem 4-2 If two angles are straight angles, then they are congruent. A ED CB F StatementsReasons Given  ABC is a straight angle and  DEF is a straight angle. Prove  ABC   DEF.

4-3 #17.Prove Theorem 4-2 If two angles are straight angles, then they are congruent. A ED CB F StatementsReasons 1.Given 2.Definition of straight angle. 3.Definition of straight angle. 4.Definition of congruent. QED Given  ABC is a straight angle and  DEF is a Straight angle. Prove  ABC   DEF. 1.  ABC is a straight angle and  DEF is a straight angle. 2.m  ABC = 180 ○ 3.m  DEF = 180 ○. Conclusion  ABC   DEF.

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 ○.

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

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 ○ 1 3 2

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 ○ 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.

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

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

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?

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?

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 ○ )

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.

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 E D B C

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 E D B C Next, given that  DBE is a straight angle, we can say that  DBC is a supplement to  EBC.

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 E D B C Next, given that  DBE is a straight angle, we can say that  DBC is a supplement to  EBC. Conclusion:  ABE   DBC

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 E D B C Next, given that  DBE is a straight angle, we can say that  DBC is a supplement to  EBC. Conclusion:  ABE   DBC 65 ○ 115 ○

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

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 E D B C Conclusion:  ABD   EBC 65 ○ 115 ○

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 E D B C Conclusion:  ABD   EBC 65 ○ 115 ○

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 E D B C 65 ○ 115 ○

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 E D B C 65 ○ 115 ○ Why 4 pairs of linear pairs?

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 E D B C 65 ○ 115 ○ Theorem 4-7: Linear pairs of angles are supplementary.

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 2 m  1 =90 ○ We can do the same for  2,  3 and 

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 E D B C 65 ○ 115 ○