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Name Class 2-6
Date Geometric Proof
Going Deeper
Essential question: How can you organize the deductive reasoning of a
geometric proof?
You will use the Angle Addition Postulate and the following
definitions to prove an important theorem about angles.
Opposite rays are two rays that have a common endpoint and form a
straight line. A linear pair of angles is a pair of adjacent angles whose
noncommon sides are opposite rays.
_​ _›
​__›
JK​
​   and ​JL​  are
In the figure,
pair of angles.
M
opposite rays; ∠MJK and ∠MJL are a linear
Recall that two angles are complementary if the sum of their measures
is 90°. Two angles are supplementary if the sum of their measures is
180°. The following theorem ties together some of the preceding ideas.
J
L
K
G-CO.3.9
1
proof
Linear Pair Theorem
If two angles form a linear pair, then they are supplementary.
M
Given: ∠MJK and ∠MJL are a linear pair of angles.
Prove: ∠MJK and ∠MJL are supplementary.
L
J
K
A Develop a plan for the proof.
© Houghton Mifflin Harcourt Publishing Company
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​__›
Since it is given that ∠MJK and ∠MJL are a linear pair of angles, JL​
​   and ​JK​  are opposite
rays. They form a straight angle. Explain why m∠MJK + m∠MJL must equal 180°.
B Complete the proof by writing the missing reasons. Choose from the following reasons.
Angle Addition Postulate
Definition of opposite rays
Substitution Property of Equality
Given
Statements
Reasons
1. ∠MJK and ∠MJL are a linear pair.
1.
2.​JL​  and JK​
​   are opposite rays.
2. Definition of linear pair
3.​JL​  and JK​
​   form a straight line.
3.
4. m∠LJK = 180°
4. Definition of straight angle
5. m∠MJK + m∠MJL = m∠LJK
5.
6. m∠MJK + m∠MJL = 180°
6.
7. ∠MJK and ∠MJL are supplementary.
7. Definition of supplementary angles
Chapter 2
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75
Lesson 6
REFLECT
1a. Is it possible to prove the theorem by measuring ∠MJK and ∠MJL in the figure and
showing that the sum of the angle measures is 180°? Explain.
1b. The proof shows that if two angles form a linear pair, then they are supplementary.
Is this statement true in the other direction? That is, if two angles are
supplementary, must they be a linear pair? Why or why not?
practice
1. You can use the Linear Pair Theorem to prove a result about
vertical angles. Complete the proof by writing the missing
statements or reasons.
Given: ∠VXW and ∠ZXY are vertical angles, as shown.
Prove: m∠VXW = m∠ZXY
Statements
V
W
X
Y
Z
Reasons
1.
2. ∠VXW and ∠ZXY are formed by
intersecting lines.
2. Definition of vertical angles
3. ∠VXW and ∠WXZ are a linear pair.
∠WXZ and ∠ZXY are a linear pair.
3. Definition of linear pair
4. ∠VXW and ∠WXZ are supplementary.
4.
5. m∠VXW + m∠WXZ = 180°
5.
6.
6. Linear Pair Theorem
7.
7. Definition of supplementary angles
8. m∠VXW + m∠WXZ = m∠WXZ + m∠ZXY
8. Transitive Property of Equality
9. m∠VXW = m∠ZXY
9.
Chapter 2
76
© Houghton Mifflin Harcourt Publishing Company
1. ∠VXW and ∠ZXY are vertical angles.
Lesson 6
2-6
Name Class Date __________________
Date Name ________________________________________
Class__________________
LESSON
Practice
2-6
Additional
Practice
Geometric Proof
Write a justification for each step.
Given: AB = EF, B is the midpoint of AC ,
and E is the midpoint of DF .
1. B is the midpoint of AC ,
and E is the midpoint of DF .
_________________________
2. AB ≅ BC , and DE ≅ EF .
_________________________
3. AB = BC, and DE = EF.
_________________________
4. AB + BC = AC, and DE + EF = DF.
_________________________
5. 2AB = AC, and 2EF = DF.
_________________________
6. AB = EF
_________________________
7. 2AB = 2EF
_________________________
8. AC = DF
_________________________
9. AC ≅ DF
_________________________
Fill in the blanks to complete the two-column proof.
10. Given: ∠HKJ
is a straight angle.
JJG
KI bisects ∠HKJ.
Prove: ∠IKJ is a right angle.
© Houghton Mifflin Harcourt Publishing Company
Proof:
Statements
Reasons
1. a._______________________________
1. Given
2. m∠HKJ = 180°
2. b. ______________________________
3. c._______________________________
3. Given
4. ∠IKJ ≅ ∠IKH
4. Def. of ∠ bisector
5. m∠IKJ = m∠IKH
5. Def. of ≅ ∠s
6. d._______________________________
6. ∠ Add. Post.
7. 2m∠IKJ = 180°
7. e. Subst. (Steps _______)
8. m∠IKJ = 90°
8. Div. Prop. of =
9. ∠IKJ is a right angle.
9. f. _______________________________
Chapter
2 Copyright © by Holt McDougal. Additions and changes 77
Lesson 6
Original content
to the original content are the responsibility of the instructor.
13
Holt McDougal Geometry
Name ________________________________________ Date __________________ Class__________________
Problem
Solving
Problem Solving
LESSON
2-6
Geometric Proof
1. Refer to the diagram of the stained-glass window and use
the given plan to write a two-column proof.
Given: ∠1 and ∠3 are supplementary.
∠2 and ∠4 are supplementary.
∠3 ≅ ∠4
Prove: ∠1 ≅ ∠2
Plan:
Use the definition of supplementary angles to write
the given information in terms of angle measures.
Then use the Substitution Property of Equality and
the Subtraction Property of Equality to conclude
that ∠1 ≅ ∠2.
2. Given: ∠1 and ∠4 are right angles.
A ∠3 ≅ ∠5
C m∠1 + m∠4 = 90°
B ∠1 ≅ ∠4
D m∠3 + m∠5 = 180°
3. Given: ∠2 and ∠3 are supplementary.
∠2 and ∠5 are supplementary.
F ∠3 ≅ ∠5
H ∠3 and ∠5 are complementary.
G ∠2 ≅ ∠5
J ∠1 and ∠2 are supplementary.
Chapter
2 Copyright © by Holt McDougal. Additions and changes to
78
Lesson 6
Original content
the original content are the responsibility of the instructor.
97
Holt McDougal Geometry
© Houghton Mifflin Harcourt Publishing Company
The position of a sprinter at the starting blocks is shown in the diagram.
Which statement can be proved using the given information? Choose the
best answer.