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4.5 Segment and Angle Proofs

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### Transcript of 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point...

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

-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

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

1. AC = AB + AB

2. AB + BC = AC

3. AB + AB = AB + BC

4. AB = BC

1. Given

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

Symmetric Property of Congruence

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

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