This document contains examples and explanations of geometric proofs involving segments, angles, and properties of equality and congruence. It includes examples proving statements about segments and angles being equal based on given information. One example proves two statements: (1) that the distance between the food court and music store is the same as the distance between the shoe store and bookstore, and (2) that this is true because the music store is halfway between the food court and shoe store. The document provides guidance on writing two-column geometric proofs and practicing proofs involving segment and angle properties.

1. 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; WX
3. If RS = WX, then ? + AB = ? + AB.
ANSWER m 1

2. 2.6

3. 2.6Example 1
Write 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 2 2. Angle Addition Postulate
3. m EBA = m 1 + m 2 3. Substitution Property of Equality
4. m 1 + m 2 = m DBC 4. Angle Addition Postulate
5. m EBA = m DBC 5. Transitive Property of Equality

4. 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

5. 2.6

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 .

7. 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.

8. 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 = AB
2
1
PROVE: a. AB = 2 AM

9. 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 = AB
2
17.b. 7. Division Property of Equality
STATEMENTS REASONS
Example 3
2. AM MB 2. Definition of midpoint

10. 2.6

11. 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

12. 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

13. 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

14. 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.

15. 2.6Exit Slip
2. Use the given information to
prove the statement.
PROVE: m 2 = 31
o
GIVEN: m 1 + m 2 = 90 ;
m 1 = 59
o
o
Statements (Reasons)
ANSWER
(Subtraction Prop. Of Eq.)2. m 2 = 90 – m 1
o
(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

16. 2.6
Homework
Pg 117-120
#3, 7, 15, 17, 21