2.5 Proving Statements about
Segments
Geometry
Standards/Objectives:
Students will learn and apply geometric
concepts.
Objectives:
• Justify statements about congruent
segments.
• Write reasons for steps in a proof.
Definitions
Theorem:
A true statement that follows as a result of
other true statements.
Two-column proof:
Most commonly used. Has numbered
statements and reasons that show the
logical order of an argument.
NOTE: Put in the Definitions/Properties/
Postulates/Theorems/Formulas portion of your notebook
• Theorem 2.1
– Segment congruence is reflexive, symmetric,
and transitive.
• Examples:
– Reflexive: For any segment AB, AB ≅ AB
– Symmetric: If AB ≅ CD, then CD ≅ AB
– Transitive: If AB ≅ CD, and CD ≅ EF, then
AB ≅ EF
Example 1: Symmetric Property of
Segment Congruence
Given: PQ ≅ XY
Prove XY ≅ PQ
Statements:
Reasons:
1. PQ ≅ XY
2. PQ = XY
1. Given
2. Definition of congruent
segments
3. Symmetric Property of
Equality
4. Definition of congruent
segments
3. XY = PQ
4. XY ≅ PQ
Example 2: Using Congruence
• Use the diagram and the given information
to complete the missing steps and reasons
in the proof.
• GIVEN: LK = 5, JK = 5, JK ≅ JL
• PROVE: LK ≅ JL
K
J
L
Statements:
1.
2.
3.
4.
5.
6.
_______________
_______________
LK = JK
LK ≅ JK
JK ≅ JL
________________
Reasons:
1.
2.
3.
4.
5.
6.
Given
Given
Transitive Property
_______________
Given
Transitive Property
Example 3: Using Segment
Relationships
• GIVEN: Q is the midpoint of PR.
• PROVE: PQ = ½ PR and QR = ½ PR.
R
Q
P
Statements:
1.
2.
3.
4.
5.
6.
7.
Q is the midpoint of PR.
PQ = QR
PQ + QR = PR
PQ + PQ = PR
2 ∙ PQ = PR
PQ = ½ PR
QR = ½ PR
Reasons:
1.
2.
Given
Definition of a midpoint
3.
Segment Addition Postulate
4.
5.
6.
7.
Substitution Property
Distributive property
Division property
Substitution
GUIDED PRACTICE
1.
for Example 1
Four steps of a proof are shown. Give the reasons
for the last two steps.
GIVEN : AC = AB + AB
PROVE : AB = BC
STATEMENT
REASONS
1. AC = AB + AB
1. Given
2. AB + BC = AC
2. Segment Addition Postulate
3. AB + AB = AB + BC
3. ?
4. AB = BC
4. ?
GUIDED PRACTICE
for Example 1
ANSWER
GIVEN : AC = AB + AB
PROVE : AB = BC
STATEMENT
REASONS
1. AC = AB + AB
1. Given
2. AB + BC = AC
2. Segment Addition Postulate
3. AB + AB = AB + BC
3. Transitive Property of Equality
4. AB = BC
4. Subtraction Property of Equality
Ex. Writing a proof:
Given: 2AB = AC
Copy or draw diagrams and label
given info to help develop proofs
A
B
C
Prove: AB = BC
Statements
Reasons
1. 2AB = AC
1. Given
2. AC = AB + BC
2. Segment addition postulate
3. 2AB = AB + BC
3. Transitive
4. AB = BC
4. Subtraction Prop.
EXAMPLE 3
Use properties of equality
GIVEN: M is the midpoint of AB .
PROVE: a. AB = 2 AM
1
b. AM = 2 AB
PROVE: a. AB = 2 AM
EXAMPLE 3
1 AB
b. AM =
2
STATEMENT
REASONS
1. M is the midpoint of AB.
1. Given
2. AM
2. Definition of midpoint
MB
3. AM = MB
3. Definition of congruent segments
4. AM + MB = AB
4. Segment Addition Postulate
5. AM + AM = AB
5. Substitution Property of Equality
a. 6. 2AM = AB
b. 7. AM = 1 AB
2
6. Addition Property
7. Division Property of Equality
Write a two-column proof
EXAMPLE 1
Write a two-column proof for
this situation
GIVEN: m 1 = m 3
PROVE: m EBA = m DBC
REASONS
STATEMENT
1. m 1 = m 3
1. Given
2. m EBA = m 3 + m 22. Angle Addition Postulate
3. m EBA = m 1 + m 23. Substitution Property of Equality
4. m 1 + m 2 = m DBC4. Angle Addition Postulate
5. m EBA = m DBC
5. Transitive Property of Equality