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 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) |
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) |
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) |
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) |