2.6 prove statements about segments and angles

2.62.6 Prove Statements about Segments and Angles Bell Thinger

Use a property of equality to complete the statement.

ANSWER AB = TU

2. If AB = CD and CD = TU, then ? .

1. If m 1 = m 3, then m 3 = ? .

ANSWER RS; WX3. If RS = WX, then ? + AB = ? + AB.

ANSWER m 1

2.6

2.6Example 1Write a two-column proof for the situation in Example 4 from Lesson 2.5.

STATEMENTS REASONS

GIVEN:m∠ 1 = m∠ 3

PROVE:m∠ EBA = m DBC

1. Given1.m 1 = m 3

2.m EBA = m 3 + m 22. Angle Addition Postulate

3.m EBA = m 1 + m 23. Substitution Property of Equality

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

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

2.6Guided Practice

GIVEN : AC = AB + AB

PROVE : AB = BC

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

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

2.6

2.6Example 2

SOLUTION

Transitive Property of Angle Congruencea.

b. Symmetric Property of Segment Congruence

Name the property illustrated by the statement.

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

b. If NK BD , then BD NK .

2.6Guided Practice

Reflexive Property of CongruenceANSWER

Symmetric Property of CongruenceANSWER

Name the property illustrated by the statement.

2. CD CD

3. If Q V, then V Q.

2.6

GIVEN: M is the midpoint of AB .

Example 3

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.

b.AM = AB21

PROVE: a. AB = 2 AM

2.6

SOLUTION

1. M is the midpoint of AB. 1. Given

3. AM = MB 3. Definition of congruent segments

4. AM + MB = AB 4. Segment Addition Postulate

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

6. 2AM = ABa. 6. Simplify

AM = AB217.b. 7. Division Property of Equality

STATEMENTS REASONS

Example 3

2. AM MB 2. Definition of midpoint

2.6

2.6 Walking down a hallway at the mall, you notice the music store is halfway between the food court and the shoe store. The shoe store is halfway between the music store and the bookstore. Prove that the distance between the entrances of the food court and music store is the same as the distance between the entrances of the shoe store and bookstore.

Shopping Mall

Example 4

2.6

SOLUTION

STEP 1 Draw and label a diagram.

STEP 2 Draw separate diagrams to show mathematical relationships.

STEP 3 State what is given and what is to be proved for the situation.Then write a proof.

Example 4

2.6

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

PROVE: AB = CD

STATEMENTS 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

Example 4

2.6Exit Slip

Reflexive Prop. Of Eq.2. ?

1. MA = TH ?

1. Copy and complete the proof.

GIVEN: MA = TH

PROVE: MT = AH

3. MA + AT = AT + TH ?

MA + AT = MT; AT + TH =AH4. ?

5. Substitution Prop. Of Eq. ?

STATEMENTS REASONS

2.

1.

3.

4.

5.

Given

AT = AT

Addition Prop. Of Eq.

Segment Add. Post.

MA + AT = MT; AT + MA=AH

GIVEN: MA = TH

PROVE: MT = AH

6. ? MT = AH Transitive Prop. Of Eq.6.

2.6Exit Slip2. Use the given information to

prove the statement.

PROVE: m 2 = 31o

GIVEN: m 1 + m 2 = 90 ;m 1 = 59

o

o

Statements (Reasons)ANSWER

(Subtraction Prop. Of Eq.)2. m 2 = 90 – m 1o

(Substitution Prop. Of Eq.)3. m 2 = 90 – 59o o

(Simplify)4. m 2 = 31o

(Given)1. m 1 + m 2 = 90 ; m 1 = 59 oo

2.6

Homework

Pg 117-120 #3, 7, 15, 17, 21