Logic and Proof Exercises

Example 1

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

Statements	Reasons
1. OG is angle bisector of ∠AOF	Given
2. ∠BOC ≅ ∠FOG	Given in figure
3. ∠COD and ∠FOG are vertical angles	Given in figure
4. ∠EOD and ∠AOG are vertical angles	Given in figure
5. ∠FOG ≅ ∠AOG	?

Hint

It almost looks like ∠AOF is split in two…

Example 2

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

Statements	Reasons
1. OG is angle bisector of ∠AOF	Given
2. ∠BOC ≅ ∠FOG	Given in figure
3. ∠COD and ∠FOG are vertical angles	Given in figure
4. ∠EOD and ∠AOG are vertical angles	Given in figure
5. ∠FOG ≅ ∠AOG	Definition of angle bisector (1)
6. ∠COD ≅ ∠FOG	?

Hint

What relationship do ∠COD and ∠FOG have?

Example 3

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

Statements	Reasons
1. OG is angle bisector of ∠AOF	Given
2. ∠BOC ≅ ∠FOG	Given in figure
3. ∠COD and ∠FOG are vertical angles	Given in figure
4. ∠EOD and ∠AOG are vertical angles	Given in figure
5. ∠FOG ≅ ∠AOG	Definition of angle bisector (1)
6. ∠COD ≅ ∠FOG	Definition of vertical angles (3)
7. ∠COD ≅ ∠AOG	?

Hint

Use ∠FOG as the middleman. Then dump it.

Example 4

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

Statements	Reasons
1. OG is angle bisector of ∠AOF	Given
2. ∠BOC ≅ ∠FOG	Given in figure
3. ∠COD and ∠FOG are vertical angles	Given in figure
4. ∠EOD and ∠AOG are vertical angles	Given in figure
5. ∠FOG ≅ ∠AOG	Definition of angle bisector (1)
6. ∠COD ≅ ∠FOG	Definition of vertical angles (3)
7. ∠COD ≅ ∠AOG	Transitive property of congruence (6 and 5)
8. ∠EOD ≅ ∠AOG	?

Hint

∠EOD and ∠AOG are right across from each other.

Example 5

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

Statements	Reasons
1. OG is angle bisector of ∠AOF	Given
2. ∠BOC ≅ ∠FOG	Given in figure
3. ∠COD and ∠FOG are vertical angles	Given in figure
4. ∠EOD and ∠AOG are vertical angles	Given in figure
5. ∠FOG ≅ ∠AOG	Definition of angle bisector (1)
6. ∠COD ≅ ∠FOG	Definition of vertical angles (3)
7. ∠COD ≅ ∠AOG	Transitive property of congruence (6 and 5)
8. ∠EOD ≅ ∠AOG	Definition of vertical angles (4)
9. ∠COD ≅ ∠EOD	?

Hint

We don't need ∠AOG anymore.

Example 6

Given: X is the midpoint of VY, X is the midpoint of WU, and WX ≅ VX.

Prove: XY ≅ XU

Statements	Reasons
1. X is the midpoint of VY	Given
2. X is the midpoint of WU	Given
3. WX ≅ VX	Given
4. ?	Definition of midpoint (1)

Hint

What does a midpoint do, exactly?

Example 7

Given: X is the midpoint of VY, X is the midpoint of WU, and WX ≅ VX.

Prove: XY ≅ XU

Statements	Reasons
1. X is the midpoint of VY	Given
2. X is the midpoint of WU	Given
3. WX ≅ VX	Given
4. VX ≅ XY	Definition of midpoint (1)
5. ?	Definition of midpoint (2)

Hint

Midpoints, take two.

Example 8

Given: X is the midpoint of VY, X is the midpoint of WU, and WX ≅ VX.

Prove: XY ≅ XU

Statements	Reasons
1. X is the midpoint of VY	Given
2. X is the midpoint of WU	Given
3. WX ≅ VX	Given
4. VX ≅ XY	Definition of midpoint (1)
5. WX ≅ XU	Definition of midpoint (2)
6. ?	Transitive property of congruence (3 and 4)

Hint

Use steps 3 and 4.

Example 9

Given: X is the midpoint of VY, X is the midpoint of WU, and WX ≅ VX.

Prove: XY ≅ XU

Statements	Reasons
1. X is the midpoint of VY	Given
2. X is the midpoint of WU	Given
3. WX ≅ VX	Given
4. VX ≅ XY	Definition of midpoint (1)
5. WX ≅ XU	Definition of midpoint (2)
6. WX ≅ XY	Transitive property of congruence (3 and 4)
7. ?	Transitive property of congruence (6 and 5)

Hint

Use steps 6 and 5.

Logic and Proof Exercises

Example 1

Example 2

Example 3

Example 4

Example 5

Example 6

Example 7

Example 8

Example 9

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