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?
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?
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.
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○