32
OTCQ Given: ABC is a straight angle and BD ABC. Prove ABD is a right angle. A B C D Statements Reasons
OTCQGiven: ABC is a straight angle and BD ABC. Prove ABD is a right angle.
A B C
DStatements Reasons
OTCQ # 100609Given: ABC is a straight angle and BD ABC. Prove ABD is a right angle.
A B C
DStatements Reasons
1. ABC is a straight angle and BD ABC.
Conclusion: ABD is a right angle
1.Given.
OTCQ # 100609Given: ABC is a straight angle and BD ABC. Prove ABD is a right angle.
A B C
DStatements Reasons
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 DBCConclusion: ABD is a right angle
1.Given.
2. Definition of perpendicular.
QED. Quo era demonstratum
Aim 4-3 How do we prove theorems about angles (part 2)?
GG 30, GG 32, GG 34
OBJECTIVE1. SWBAT prove angle theorems.2. SWBAT recall some basic
definitions.
Prove Theorem 4-2 If two angles are straight angles, then they are congruent.
A
ED
CB
F
Statements Reasons1. 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.
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.
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: If 2 angles are complements of the same angle then they are congruent.
Why?
Theorem: If 2 angles are complements of the same angle then they are congruent.
Why?Givenm 1= 45○
m 2= 45○
m 3= 45○
13
2
Theorem: If 2 angles are complements of the same angle then they are congruent.
Why?Givenm 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: If 2 angles are congruent then their complements are congruent.
Why?
Theorem: If 2 angles are congruent then their complements are congruent.
Why?Givenm 1= 30○
m 2= 30○ 1
3
24
Theorem: 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
32
4
Theorem: 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
32
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
32
4
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.
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 E
D
B
C
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 E
D
B
CNext, given that DBE is a straight angle, we can say that DBC is a supplement to EBC.
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 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: 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: If 2 angles are congruent then their supplements are congruent.
Why?
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 E
D
B
C
Conclusion: ABD EBC
65○
65○
115○
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 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: Linear pairs of angles are supplementary.
Theorem: If 2 lines intersect to form congruent adjacent angles, then they are perpendicular.
1
3
2
4
Theorem: 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○
34
G
F
E
H
1 2
D
C
B
AGiven: ADC EHG 1 4
Prove: 2 3Statements Reasons
Recall Properties of Equality
1) Reflexive: a = a2) Symmetric: If a = b then b = a.3) Transitive:
If a = b and b = c, then a = c.4) Substitution: If a = b, then a can be replaced by b.
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