1. 4.5 Segment and Angle Proofs

2. Basic geometry symbols you need to know
Word(s)
Symbol
Definition
Point A
Line AB
Line Segment AB
Ray
Angle ABC
Measure of angle ABC
Congruent

3. Substitution properties
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. (duh) 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.

4. 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

5. Segment Addition Postulate -if B is between A and C, then AB + BC = AC
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….

6. 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 -

7. 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

8. Write a two-column proof
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
STATEMENT
REASONS 1.
m∠ 1 = m∠ 3 1.
Given 2.
m∠ EBA = m∠ m∠ 2 2.
Angle Addition Postulate 3.
m∠ EBA = m∠ m∠ 2 3.
Substitution Property of Equality 4.
m∠ m∠ 2 = m∠ DBC 4.
Angle Addition Postulate 5.
m∠ EBA = m∠ DBC 5.
Transitive Property of Equality

9. GUIDED PRACTICE for Example 1 GIVEN : AC = AB + AB PROVE : AB = BC
ANSWER
STATEMENT
REASONS 1.
AC = AB + AB 1.
Given 2.
AB + BC = AC 2.
Segment Addition Postulate 3.
AB + AB = AB + BC 3.
Transitive Property of Equality 4.
AB = BC 4.
Subtraction Property of Equality

10. 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 Congruence a. b.
Symmetric Property of Segment Congruence

11. GUIDED PRACTICE
for Example 2
Name the property illustrated by the statement.
2. CD CD
Reflexive Property of Congruence
ANSWER
3. If Q V, then V Q.
Symmetric Property of Congruence
ANSWER

12. 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 B A

13. 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 = AB 2 1

14. Use properties of equality
EXAMPLE 3
Use properties of equality
STATEMENT
REASONS 1.
M is the midpoint of AB. 1.
Given 2.
AM MB 2.
Definition of midpoint 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 = AB a. 6.
Distributive Property
AM = AB 2 1 7. b. 7.
Division Property of Equality

15. EXAMPLE 4
Solve a multi-step problem
Shopping Mall
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.

16. EXAMPLE 4
Solve a multi-step problem
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.

17. Solve a multi-step problem
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.
AB BC 2.
Definition of midpoint 3.
BC CD 3.
Definition of midpoint 4.
AB CD 4.
Transitive Property of Congruence 5.
AB = CD 5.
Definition of congruent segments