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Logic and Proof

Logic and Proof

Congruence, Equality, and Geometry Exercises

Example 1

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

StatementsReasons
1. OG is angle bisector of ∠AOFGiven
2. ∠BOC ≅ ∠FOGGiven in figure
3. ∠COD and ∠FOG are vertical anglesGiven in figure
4. ∠EOD and ∠AOG are vertical anglesGiven in figure
5. ∠FOG ≅ ∠AOG?

Example 2

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

StatementsReasons
1. OG is angle bisector of ∠AOFGiven
2. ∠BOC ≅ ∠FOGGiven in figure
3. ∠COD and ∠FOG are vertical anglesGiven in figure
4. ∠EOD and ∠AOG are vertical anglesGiven in figure
5. ∠FOG ≅ ∠AOGDefinition of angle bisector (1)
6. ∠COD ≅ ∠FOG?

Example 3

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

StatementsReasons
1. OG is angle bisector of ∠AOFGiven
2. ∠BOC ≅ ∠FOGGiven in figure
3. ∠COD and ∠FOG are vertical anglesGiven in figure
4. ∠EOD and ∠AOG are vertical anglesGiven in figure
5. ∠FOG ≅ ∠AOGDefinition of angle bisector (1)
6. ∠COD ≅ ∠FOGDefinition of vertical angles (3)
7. ∠COD ≅ ∠AOG?

Example 4

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

StatementsReasons
1. OG is angle bisector of ∠AOFGiven
2. ∠BOC ≅ ∠FOGGiven in figure
3. ∠COD and ∠FOG are vertical anglesGiven in figure
4. ∠EOD and ∠AOG are vertical anglesGiven in figure
5. ∠FOG ≅ ∠AOGDefinition of angle bisector (1)
6. ∠COD ≅ ∠FOGDefinition of vertical angles (3)
7. ∠COD ≅ ∠AOGTransitive property of congruence (6 and 5)
8. ∠EOD ≅ ∠AOG?

Example 5

Given: OG is an angle bisector of ∠AOF.

Prove: ∠COD ≅ ∠EOD

StatementsReasons
1. OG is angle bisector of ∠AOFGiven
2. ∠BOC ≅ ∠FOGGiven in figure
3. ∠COD and ∠FOG are vertical anglesGiven in figure
4. ∠EOD and ∠AOG are vertical anglesGiven in figure
5. ∠FOG ≅ ∠AOGDefinition of angle bisector (1)
6. ∠COD ≅ ∠FOGDefinition of vertical angles (3)
7. ∠COD ≅ ∠AOGTransitive property of congruence (6 and 5)
8. ∠EOD ≅ ∠AOGDefinition 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

StatementsReasons
1. X is the midpoint of VYGiven
2. X is the midpoint of WUGiven
3. WXVXGiven
4. ?Definition of midpoint (1)

Example 7

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

Prove: XYXU

StatementsReasons
1. X is the midpoint of VYGiven
2. X is the midpoint of WUGiven
3. WXVXGiven
4. VXXYDefinition 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

StatementsReasons
1. X is the midpoint of VYGiven
2. X is the midpoint of WUGiven
3. WXVXGiven
4. VXXYDefinition of midpoint (1)
5. WXXUDefinition 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

StatementsReasons
1. X is the midpoint of VYGiven
2. X is the midpoint of WUGiven
3. WXVXGiven
4. VXXYDefinition of midpoint (1)
5. WXXUDefinition of midpoint (2)
6. WXXYTransitive property of congruence (3 and 4)
7. ?Transitive property of congruence (6 and 5)