4.5 Segment and Angle Proofs

Basic geometry symbols you need to knowWord(s) Symbol Definition

Point A

Line AB

Line Segment AB

Ray

Angle ABC

Measure of angle ABC

Congruent

Supplementary Angles

Complementary Angles

Congruent Angles

Substitution properties If you prove 2 parts are congruent, they can substitute

Reflexive properties The same part is equal to the same part. AB=AB

Transitive properties If part 1 = part 2, and part 2 = part 3, then 1 = 3

Symmetric properties If A = B then B = A.

Vocabulary Proof – a logical argument that shows a

statement is true

Two – column proof – numbered statements in one column, corresponding reason in other

Statement Reasons

Postulate – a rule that is accepted without proof. Write forwards and backwards – reverse it and it‘s still true

Segment Addition Postulate

-if B is between A and C, then AB + BC = AC

converse….. Reverse it….

- if AB + BC = AC, then B is between A and C

Draw it….

Angle Addition Postulate – - If P is the interior (inside) of <RST then the measure

of < RST is equal to the sum of the measures of <RSP and <PST.

Draw it -

Theorem – a statement that can be proven Theorem 4.1 – Congruence of Segments Reflexive –

Symmetric –

Transitive –

Theorem 4.2 – Congruence of Angles Reflexive

Symmetric

Transitive

EXAMPLE 1 Write a two-column proof

Write a two-column proof for the situation in Example 4 from Lesson 2.5.

GIVEN:m∠1 = m∠3

PROVE:m∠EBA = m∠DBC

1.m∠1 = m∠32.m∠EBA = m∠3 + m∠23.m∠EBA = m∠1 + m∠2

1. Given2. Angle Addition Postulate

3. Substitution Property of Equality

STATEMENT REASONS

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

5.m∠EBA = m∠DBC 5. Transitive Property of Equality

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

EXAMPLE 2 Name the property shown

Name the property illustrated by the statement.

a. If R T and T P, then R P.

b. If NK BD , then BD NK .

SOLUTION

Transitive Property of Angle Congruencea.

b. Symmetric Property of Segment Congruence

GUIDED PRACTICE for Example 2

2. CD CD

3. If Q V, then V Q.

Reflexive Property of Congruence

ANSWER

Symmetric Property of Congruence

ANSWER

Name the property illustrated by the statement.

Solving for x. Based on the properties learned, if you know 2

“parts” are congruent, you set them equal to each other and solve.

m<A=2x+15, m<B=4x-3 2x+15=4x-3 15=2x-3 18=2x 9=x 2(9)+15=18+15=33 A B

A B∠ ∠

EXAMPLE 3 Use properties of equality

Prove this property of midpoints: If you know that M is the midpoint of AB ,prove that AB is two times AM and AM is one half of AB.

GIVEN: M is the midpoint of AB .

PROVE: a. AB = 2 AM

b.AM = AB21

STATEMENT REASONS

EXAMPLE 3 Use properties of equality

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

6. 2AM = ABa.

AM = AB217.b.

6. Distributive Property

7. Division Property of Equality

EXAMPLE 4 Solve a multi-step problem

GIVEN: B is the midpoint of AC .C is the midpoint of BD .

PROVE: AB = CD

STATEMENT REASONS

1. B is the midpoint of AC .C is the midpoint of BD .

1. Given

2. Definition of midpoint2. AB BC

3. BC CD 3. Definition of midpoint

5. AB = CD

4. AB CD 4. Transitive Property of Congruence

5. Definition of congruent segments