# Congruence, Equality, and Geometry 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 ?

### 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 ?

### 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 ?

### 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 ?

### 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 ?

### Example 6

Given: X is the midpoint of VY, X is the midpoint of WU, and WXVX.

Prove: XYXU

 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)

### Example 7

Given: X is the midpoint of VY, X is the midpoint of WU, and WXVX.

Prove: XYXU

 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)

### Example 8

Given: X is the midpoint of VY, X is the midpoint of WU, and WXVX.

Prove: XYXU

 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)

### Example 9

Given: X is the midpoint of VY, X is the midpoint of WU, and WXVX.

Prove: XYXU

 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)