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# Logic and Proof

# 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*.

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 in figure |

4. ? | Definition of midpoint (1) |

### Example 7

Given: *X* is the midpoint of *VY*, *X* is the midpoint of *WU*.

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 in figure |

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*.

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 in figure |

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*.

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 in figure |

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) |