1. 2.5 Proving Statements about Segments
Geometry

2. Standards/Objectives:
Students will learn and apply geometric concepts.
Objectives:
Justify statements about congruent segments.
Write reasons for steps in a proof.

3. 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.

4. NOTE: Put in the Definitions/Properties/ Postulates/Theorems/Formulas portion of your notebook
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

5. Example 1: Symmetric Property of Segment Congruence
Given: PQ ≅ XY
Prove XY ≅ PQ
Statements:
PQ ≅ XY
PQ = XY
XY = PQ
XY ≅ PQ
Reasons:
Given
Definition of congruent segments
Symmetric Property of Equality

6. 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

7. _______________ LK = JK LK ≅ JK JK ≅ JL ________________ Given
Statements: Reasons:
_______________
LK = JK
LK ≅ JK
JK ≅ JL
________________
Given
Transitive Property
_______________

8. Example 3: Using Segment Relationships
GIVEN: Q is the midpoint of PR.
PROVE: PQ = ½ PR and QR = ½ PR.

9. Definition of a midpoint Substitution Property Distributive property
Statements: Reasons:
Q is the midpoint of PR.
PQ = QR
PQ + QR = PR
PQ + PQ = PR
2 ∙ PQ = PR
PQ = ½ PR
QR = ½ PR
Given
Definition of a midpoint
Segment Addition Postulate
Substitution Property
Distributive property
Division property
Substitution

10. GUIDED PRACTICE for Example 1
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. ?

11. GUIDED PRACTICE for Example 1 ANSWER GIVEN : AC = AB + AB
PROVE : AB = BC
ANSWER 1.
AC = AB + AB 2.
AB + BC = AC 3.
AB + AB = AB + BC 4.
AB = BC
Given
Segment Addition Postulate
Transitive Property of Equality
Subtraction Property of Equality
STATEMENT
REASONS

12. Ex. Writing a proof: Given: 2AB = AC Prove: AB = BC A B C 2AB = AC
Copy or draw diagrams and label given info to help develop proofs
Ex. Writing a proof:
Given: 2AB = AC
Prove: AB = BC A B C
Statements
Reasons
2AB = AC
AC = AB + BC
2AB = AB + BC
AB = BC
Given
Segment addition postulate
Transitive
Subtraction Prop.

13. EXAMPLE 3
Use properties of equality
GIVEN:
M is the midpoint of AB .
PROVE: a.
AB = 2 AM b.
AM = AB 2 1

14. a. EXAMPLE 3 STATEMENT REASONS 1. M is the midpoint of AB. 1. 2. AM MB
PROVE: a.
AB = 2 AM
EXAMPLE 3 b.
AM = AB 2 1
STATEMENT
REASONS 1.
M is the midpoint of AB. 1.
Given 2.
AM MB 2.
Definition of midpoint 3.
Definition of congruent segments 3.
AM = MB 4.
Segment Addition Postulate 4.
AM + MB = AB 5.
Substitution Property of Equality 5.
AM + AM = AB 6.
Addition Property 6.
2AM = AB a.
AM = AB 2 1 7. b. 7.
Division Property of Equality

15. 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
STATEMENT
REASONS 1.
m∠ 1 = m∠ 3 1.
Given 2.
m∠ EBA = m∠ m∠ 2 2.
Angle Addition Postulate 3.
m∠ EBA = m∠ m∠ 2 3.
Substitution Property of Equality 4.
m∠ m∠ 2 = m∠ DBC 4.
Angle Addition Postulate 5.
m∠ EBA = m∠ DBC 5.
Transitive Property of Equality