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

15
4.5 Segment and Angle Proofs

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

Page 1: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

4.5 Segment and Angle Proofs

Page 2: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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

Page 3: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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.

Page 4: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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

Page 5: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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….

Page 6: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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 -

Page 7: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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

Page 8: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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

Page 9: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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

Page 10: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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

Page 11: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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.

Page 12: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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∠ ∠

Page 13: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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

Page 14: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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

Page 15: 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point A Line AB Line Segment AB Ray Angle ABC Measure of.

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