OTCQWhat 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

ObjectivesSWBAT 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

Statements Reasons

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

Statements Reasons

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○

13

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○

13

2

m1+ m 2= 90○ , hence 2 is the complement of 1.m1+ 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○ 1

3

2

4

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

Why?Givenm 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?

1

3

2

4

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

Why?Givenm 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?

1

3

2

4

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

Why? 3 4Givenm 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○)

1

3

2

4

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

CNext, 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

CNext, 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

CNext, given that DBE is a straight angle, we can say that DBC is a supplement to EBC.

Conclusion: ABE DBC

65○

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○

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○

65○

115○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○

65○

115○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○

65○

115○115○

Why 4 pairsof 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○

65○

115○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.

1

3

2

4

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

Since m1 + m 2 =180○ and 1 2, we may substitute to say

m 1 + m 1 =180○ and then2 m 1 =180○ and then

2 m 1 =180○ and then 2 2m 1 =90○

We can do the same for 2, 3 and 4

1

3

2

4

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○

65○

115○115○