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Proving Angles Congruent
During this lesson, you will:
 Determine and apply conjectures about
angle relationships
 Prove and apply theorems about angles
Mrs. McConaughy
Geometry
1
Part I: Discovering Angle
Relationships
Mrs. McConaughy
Geometry
2
Definitions: Special Angle
Pairs
complementary angles
Two angles are ___________________
if their measures add up to 90.
supplementary angles
Two angles are ___________________
if their measures add up to 180.
Mrs. McConaughy
Geometry
3
Vocabulary Review: Pairs of Angles
Formed By Intersecting Lines
Opposite (non-adjacent) angles, formed by
intersecting lines, which share a common
vertex and whose sides are opposite rays
vertical angles
are called ______________.
Adjacent angles formed by intersecting
lines which share a common vertex, a
common side, and with one side formed by
linear pairs
opposite rays are called ____________.
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Geometry
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Given the following diagram,
identify all vertical angle pairs:
1

4

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
2
3
∠ 2
1 & ∠ 4
3
Geometry
5
Given the following diagram,
identify all linear pairs of angles:
2
8
4
6
∠ 8
2 & ∠ 2
4
6
4
6
8
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Geometry
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Investigative Results:
If two angles are vertical angles, then
congruent
the angles are _________.
(VERTICAL ANGLES CONJECTURE)
If two angles are a linear pair of
angles, then the angles are
supplementary
180
______________ (____).
(LINEAR PAIR CONJECTURE)
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Geometry
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If two angles are equal and
supplementary, what must be true
of the two angles?
If two angles are both equal in
measure and supplementary, then
90
each angle measures ____.
(EQUAL SUPPLEMENTS CONJECTURE)
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Geometry
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Examples: Use your conjectures to find
the measure of each lettered angle.
Example A
a
a
b
c
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Example B
70
30
Geometry
b
c
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Examples: Use your conjectures to a.
find the value of the variable.
EXAMPLE C
EXAMPLE D
(2x – 6)
(3y + 20)
Vertical
Angles
Are Congruent
Linear Pairs Are
Supplementary
(3x + 31)
(5y – 16)
5y – 16 = 3y + 20
5y = 3y + 36
5y = 3y + 36
3x + 31 + 2x – 6 = 180
5x + 25 = 180
2y = 36
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y = 18
Geometry
5x = 155
x = 31
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Homework Assignment:
Discovering Angle Relationships
WS 1-5 all, plus select
problems from text.
Mrs. McConaughy
Geometry
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Part 2: Proving and
Applying Theorems
About Angles
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Geometry
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Congruent Supplements
Theorem
If two angles are supplements of
congruent angles, then the two angles
are congruent.
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Geometry
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Given: ∠A supp ∠B; ∠C supp ∠D; ∠B  ∠C
Prove: ∠A  ∠D
STATEMENT
REASON
1. ∠A supp ∠B; ∠C supp ∠D 1. Given.
m
2. ∠A + m ∠B = 180; m ∠C + m ∠D = 180 2.Def. of supp. ∠’s
m ∠A + m ∠B = m ∠C + m ∠D
3.. Substitution Prop. of =
4. ∠B  ∠C
4. Given.
5. m ∠B = m ∠C
5. Def. of 
6. m ∠A = m ∠ D
7. ∠A  ∠C
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6. Subtraction Prop. of =
7. Def. of 
Geometry
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Vocabulary: Corollary
corollary of a theorem is a
A _________
theorem whose proof contains only a
few additional statements in addition to
the original proof.
EXAMPLE:
If two angles are supplements of
the same angle, then the two angles
are congruent.
Mrs. McConaughy
Geometry
15
Congruent Complements
Theorem
If two angles are complements of
congruent angles, then the two angles
are congruent.
COROLLARY: If two angles are
complements of the same angle, then
the two angles are congruent.
Mrs. McConaughy
Geometry
16
Given: ∠A comp . ∠B; ∠C comp. ∠D; ∠B  ∠C
Prove: ∠A  ∠D
STATEMENT
REASON
1. ∠A comp ∠B; ∠C comp ∠D
1. Given.
m2.∠A + m ∠B = 90; m ∠C + m ∠D = 90 2. Def. of supp. ∠’s
m3.∠A + m ∠B = m ∠C + m ∠D
3. Substitution Prop. of =
4. Given.
4.∠B  ∠C
5. m ∠B = m ∠C
5. Def. of 
6 (-) Prop. of =
7. Def. of 
6. m ∠A = m ∠ D
7. ∠A  ∠C
Mrs. McConaughy
Geometry
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Vertical Angles
Theorem
Vertical angles are congruent.
Given: ∠ 1 and ∠ 3 are
vertical angles
1
3
2
Prove: ∠ 1  ∠ 3
Mrs. McConaughy
Geometry
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Given: ∠ 1 and ∠ 3 are vertical angles
Prove: ∠ 1  ∠ 3
STATEMENT
REASON
∠1.1 and ∠ 3 are vertical angles
2.∠ 1 and ∠2 are a linear pair
1. Given.
Def. of linear pair
2.
3.∠ 2 and ∠3 are a linear pair
4. ∠1 supp ∠2; ∠3 supp ∠2
3.Def. of linear pair
Linear pairs are supp.
4.
5. ∠ 1  ∠ 3
5.
Supp. of same
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Geometry
∠ 
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Theorem
All right angles are congruent.
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Geometry
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Final Checks for Understanding
In the following exercises, ∠ 1 and ∠ 3
are a linear pair, ∠ 1 and ∠ 4 are a
linear pair, and ∠ 1 and ∠ 2 are vertical
angles. Is the statement true?
a.
b.
c.
f.
∠ 1  ∠ 3
b. ∠ 1  ∠ 2
c. ∠ 1  ∠ 4
d. ∠ 3  ∠ 2
e. ∠ 3  ∠ 4
m∠ 2 + m ∠ 3 = 180
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Geometry
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Homework Assignment
Pages 100-101: 10-18 all. 32-35
all. Prove: 19 & 35 all.
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Geometry
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