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 ≅ XYProve XY ≅ PQ

Statements:

1. PQ ≅ XY2. PQ = XY

3. XY = PQ

4. XY ≅ PQ

Reasons:

1. Given2. Definition of congruent

segments3. Symmetric Property of

Equality4. Definition of congruent

segments

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≅

1. _______________

2. _______________

3. LK = JK

4. LK ≅ JK5. JK ≅ JL6. ________________

1. Given

2. Given

3. Transitive Property

4. _______________

5. Given

6. Transitive Property

Statements: Reasons:

Example 3: Using Segment Relationships

• GIVEN: Q is the midpoint of PR.

• PROVE: PQ = ½ PR and QR = ½ PR.

1. Q is the midpoint of PR.2. PQ = QR3. PQ + QR = PR4. PQ + PQ = PR5. 2 ∙ PQ = PR6. PQ = ½ PR7. QR = ½ PR

1. Given2. Definition of a midpoint3. Segment Addition

Postulate

4. Substitution Property5. Distributive property6. Division property7. Substitution

Statements: Reasons:

GUIDED PRACTICE for Example 1

GIVEN : AC = AB + AB

PROVE : AB = BC

1. Four steps of a proof are shown. Give the reasons for the last two steps.

1. AC = AB + AB

2. AB + BC = AC

3. AB + AB = AB + BC

4. AB = BC

1. Given

2. Segment Addition Postulate

STATEMENT REASONS

3. ?

4. ?

GUIDED PRACTICE for Example 1

GIVEN : AC = AB + AB

PROVE : AB = BC

ANSWER

1. AC = AB + AB

2. AB + BC = AC

3. AB + AB = AB + BC

4. AB = BC

1. Given

2. Segment Addition Postulate

3. Transitive Property of Equality

4. Subtraction Property of Equality

STATEMENT REASONS

Ex. Writing a proof:

Given: 2AB = AC

Prove: AB = BCA B C

Copy or draw diagrams and label given info to help develop proofs

Statements Reasons

1. 2AB = AC

2. AC = AB + BC

3. 2AB = AB + BC

4. AB = BC

1. Given

2. Segment addition postulate

3. Transitive

4. Subtraction Prop.

EXAMPLE 3 Use properties of equality

GIVEN: M is the midpoint of AB .

PROVE: a. AB = 2 AM

b.AM = AB21

STATEMENT REASONS

EXAMPLE 3

1. M is the midpoint of AB.

2. AM MB

3. AM = MB

4. AM + MB = AB

1. Given

2. Definition of midpoint

3. Definition of congruent segments

4. Segment Addition Postulate

5. AM + AM = AB 5. Substitution Property of Equality

PROVE: a. AB = 2 AM

b. AM = AB21

6. 2AM = ABa.

AM = AB217.b.

6. Addition Property

7. Division Property of Equality

EXAMPLE 1Write a two-column proof

Write a two-column proof for this situation

GIVEN:m∠1 = m∠3

PROVE:m∠EBA = m∠DBC

1.m∠1 = m∠3

2.m∠EBA = m∠3 + m∠2

3.m∠EBA = m∠1 + m∠2

1. Given

2. Angle Addition Postulate

3. Substitution Property of Equality

STATEMENT REASONS

5.m∠EBA = m∠DBC

4.m∠1 + m∠2 = m∠DBC4. Angle Addition Postulate

5. Transitive Property of Equality