2.6 Prove Statements about Segments and Angles
Goal  Write proofs using geometric theorems.
Your Notes
VOCABULARY
Proof
A proof is a logical argument that shows a statement is true.
Two-column proof
A two-column proof has numbered statements and corresponding reasons that show an
argument in logical order.
Theorem
A theorem is a statement that can be proven.
Example 1
Write a two-column proof
Use the diagram to prove m1 = m4.
Given m2 = m3, mAXD = mAXC
Prove m1= m4
Writing a two-column
proof is a formal way of
organizing your reasons to
show a statement is true.
Statements
Reasons
1. mAXC = mAXD
1. _Given_
2. mAXD = m_1_ + m_2_
2. Angle Addition Postulate
3. mAXC = m_3_ + m_4_
3. Angle Addition Postulate
4. ml + m2= m3 + m4
4. _Substitution Property of Equality_
5. m2 = m3
5. _Given_
6. ml + m_3_ = m3 + m4
6. Substitution Property of Equality
7. ml = m4
7. _Subtraction Property of Equality_
Your Notes
THEOREM 2.1 CONGRUENCE OF SEGMENTS
Segment congruence is reflexive, symmetric, and transitive.
Reflexive
For any segment AB, AB  AB.
Symmetric
If AB  CD , then CD  AB .
Transitive
If AB  CD and CD  EF , then AB  EF .
THEOREM 2.2 CONGRUENCE OF ANGLES
Angle congruence is reflexive, symmetric, and transitive.
Reflexive
For any angle A, _A  A_.
Symmetric
If A  B, then _B  A_.
Transitive
If A  B and B  C, then _A  C_.
Example 2
Name the property shown
Name the property illustrated by the statement.
If 5  3, then 3  5.
_Symmetric Property of Angle Congruence_
Checkpoint Complete the following exercises.
1. Three steps of a proof are shown. Give the reasons for the last two steps.
Given BC = AB
Prove AC = AB + AB
Statements
1. BC = AB
2. AC = AB + BC
Reasons
1.Given
2._Segment Addition Postulate_
3. AC =AB + AB
3._Substitution Property of Equality_
2. Name the property illustrated by the statement. If H  T and T  B, then H
 B.
Transitive Property of Angle Congruence
Your Notes
Example 3
Use properties of equality

If you know that BD bisects ABC, prove that mABC is two times ml.

Given BD bisects ABC.
Prove mABC = 2  ml
Before writing a proof,
organize your reasoning by
copying or drawing a diagram
for the situation described.
Then identify the GIVEN and
PROVE statements.
Statements
Reasons

1. BD bisects ABC.
2. _1  2_
1. _Given_
3. _m1 = m2_
3. Definition of congruent
angles
4. _Angle Addition Postulate_
4. ml + m2 = mABC
5. m1 = m_1_ = mABC
6. _2  m1 = mABC_
2. Definition of angle bisector
5. Substitution Property of
Equality
6. Distributive Property
CONCEPT SUMMARY: WRITING A TWO-COLUMN PROOF
Proof of the Symmetric Property of Segment Congruence
Copy or draw diagrams
and label information to
help develop proofs.
Given AB  CD
Prove CD  AB
Statement
based on facts
that you know
or conclusions
from deductive
reasoning
Statement
1. AB  CD
Reasons
1. _Given_
2. _AB = CD_
3. _CD = AB_
4. CD  AB
2. Definition of congruent segments
3. Symmetric Property of Equality
4. Definition of congruent segments
The number of Remember to
statements will give a reason for
very.
the last
statements
Definitions,
postulates, or
proven theorems
that allow you to
state the
corresponding
statement.
Your Notes
Example 4
Solve a multi-step problem
Interstate There are two exits between rest areas on a stretch of interstate. The Rice exit
is halfway between rest area A and the Mason exit. The distance between rest area B and
the Mason exit is the same as the distance between rest area A and the Rice exit. Prove
that the Mason exit is halfway between the Rice exit and rest area B.
Solution
Step 1 Draw a diagram.
Step 2 Draw diagrams showing relationships.
Step 3 Write a proof.
Given R is the midpoint of, AM , MB = AR .
Prove M is the midpoint of RB .
Statements
Reasons
1. R is the midpoint of AM , MB = AR . 1. _Given_
2.
AR ______
RM
2. Definition of midpoint
3. _AR = RM_
3. Definition of congruent segments
4. MB = RM
4. _Transitive Property of Congruence_
5. MB ______
RM
6. M is the midpoint of RB.
5. Definition of congruent segments
6. _Definition of midpoint_
Checkpoint Complete the following exercise.
3. In Example 4, there are rumble strips halfway between the Rice and Mason exits.
What other two places are the same distance from the rumble strips?
Rest area A and rest area B
Homework
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