Mrs. McConaughy Geometry 1
Proving Angles Congruent
During this lesson, you will: Determine and apply conjectures
about angle relationships Prove and apply theorems about
angles
Mrs. McConaughy Geometry 2
Part I: Discovering Angle Relationships
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Definitions: Special Angle Pairs
Two angles are ___________________ if their measures add up to 90.
Two angles are ___________________ if their measures add up to 180.
complementary angles
supplementary angles
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Vocabulary Review: Pairs of Angles Formed By Intersecting
LinesOpposite (non-adjacent) angles, formed
by intersecting lines, which share a common vertex and whose sides are opposite rays are called ______________.
Adjacent angles formed by intersecting lines which share a common vertex, a common side, and with one side formed by opposite rays are called ____________.
vertical angles
linear pairs
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Given the following diagram, identify all vertical angle
pairs:
∠ 1 & ∠ 3
12
34
∠ 2 & ∠ 4
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Given the following diagram, identify all linear pairs of
angles:
2
4
68
∠ 2 & ∠ 4 ∠ 4 & ∠ 6 ∠ 6 & ∠ 8 ∠ 8 & ∠ 2
Mrs. McConaughy Geometry 7
supplementary 180
Investigative Results:
If two angles are vertical angles, then
the angles are _________. (VERTICAL ANGLES CONJECTURE)
If two angles are a linear pair of angles, then the angles are ______________ (____).
(LINEAR PAIR CONJECTURE)
congruent
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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 each angle measures ____.
(EQUAL SUPPLEMENTS CONJECTURE)
90
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Examples: Use your conjectures to find the measure of each lettered angle.
Example B
a
b c
Example A
a b
c 70 30
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Examples: Use your conjectures to a. find the value of the
variable.EXAMPLE C
(3y + 20)
(5y – 16)
EXAMPLE D
(2x – 6)
(3x + 31)
5y – 16 = 3y + 20
5y = 3y + 365y = 3y + 36
2y = 36
y = 18
Vertical Angles Are Congruent
Linear Pairs Are Supplementary
3x + 31 + 2x – 6 = 1805x + 25 = 180
5x = 155
x = 31
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Homework Assignment:
Discovering Angle Relationships WS 1-5 all, plus select problems from text.
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Part 2: Proving and Applying Theorems
About Angles
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Congruent Supplements TheoremIf two angles are supplements of congruent angles, then the two angles are congruent.
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Given: ∠A supp ∠B; ∠C supp ∠D; ∠B ∠C Prove: ∠A ∠DSTATEMENT REASON
1. 1.
2. 2.
3.
4. 4.
5. 5.
6. 6.
7. 7.
∠A supp ∠B; ∠C supp ∠D
Given.
Def. of supp. ’s∠m A + m B = 180; m C + m D = 180∠ ∠ ∠ ∠
. Substitution Prop. of =
∠B ∠C Given.
m ∠B = m ∠C
Subtraction Prop. of =∠A ∠C
Def. of
m ∠A + m ∠B = m ∠C + m ∠D
m ∠A = m ∠ D
Def. of
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Vocabulary: Corollary
A _________ of a theorem is 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.
corollary
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Congruent Complements TheoremIf 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.
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Given: ∠A comp . ∠B; ∠C comp. ∠D; ∠B ∠C Prove: ∠A ∠D
STATEMENT REASON
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6
7. 7.
∠A comp ∠B; ∠C comp ∠D
Given.
m ∠A + m ∠B = 90; m ∠C + m ∠D = 90
Def. of supp. ’s∠
m ∠A + m ∠B = m ∠C + m ∠D
Substitution Prop. of =
∠B ∠C Given.
m ∠B = m ∠C Def. of m ∠A = m ∠ D (-) Prop. of =
∠A ∠C Def. of
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Vertical Angles TheoremVertical angles are congruent.
Given: ∠ 1 and ∠ 3 are
vertical angles 1
3 2
Prove: ∠ 1 ∠ 3
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Given: ∠ 1 and ∠ 3 are vertical anglesProve: ∠ 1 ∠ 3
STATEMENT REASON
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
∠ 1 and ∠ 3 are vertical angles
Given.
∠ 1 and ∠2 are a linear pair
Def. of linear pair
∠ 2 and ∠3 are a linear pair
Def. of linear pair∠1 supp ∠2; ∠3 supp ∠2
Linear pairs are supp.
∠ 1 ∠ 3 Supp. of same ∠
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Theorem
All right angles are congruent.
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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. ∠ 1 ∠ 3 b. ∠ 1 ∠ 2 b. c. ∠ 1 ∠ 4 d. ∠ 3 ∠ 2 c. e. ∠ 3 ∠ 4 f. m∠ 2 + m ∠ 3 = 180
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Homework Assignment
Pages 100-101: 10-18 all. 32-35 all. Prove: 19 & 35 all.