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Even More Geometric Proofs
If two angles form a linear pair, then they are
supplementary.
If two angles are supplementary to the same angle,
then the two angles are congruent.
Linear Pair Theorem
Congruent Supplements Theorem
Right Angle Congruence Theorem
All right angles are congruent.
Congruent Complements Theorem
If two angles are complementary to the same angle,
then the two angles are congruent.
Vertical Angles Theorem
Vertical angles are congruent.
⃗⃗⃗⃗⃗ bisects ∠AOC, ⃗⃗⃗⃗⃗
1. Given: 𝑂𝐵
𝑂𝐸 bisects ∠DOF, ∠AOB ≅ ∠DOE
Prove: ∠EOF ≅ ∠BOC
Statements
⃗⃗⃗⃗⃗ bisects ∠AOC, ⃗⃗⃗⃗⃗
1. 𝑂𝐵
𝑂𝐸 bisects ∠DOF
2. ∠AOB ≅ ∠BOC, ∠DOE ≅ ∠EOF
3. ∠AOB ≅ ∠DOE
4. ∠AOB ≅ ∠EOF
5. ∠EOF ≅ ∠BOC
Reasons
1.
2.
3.
4.
5.
Choices:
Definition of Angle Bisector
Given
Transitive Property of Congruence
2. Given: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐶𝐷
̅̅̅̅ ≅ 𝐵𝐷
̅̅̅̅
Prove: 𝐴𝐶
Choices:
Addition Property
Definition of Congruent Segments
Given
Reflexive Property
Segment Addition Postulate
Substitution Property
3. Given: RS = UV, ST = TU
̅̅̅̅ ≅ ̅̅̅̅
Prove: 𝑅𝑇
𝑇𝑈
Statements
1. RS = UV, ST = TU
2. RS + ST = RT, TU + UV = TV
3. RS + ST = TU + UV
4. RT = TV
̅̅̅̅ ≅ 𝑇𝑈
̅̅̅̅
5. 𝑅𝑇
Reasons
1.
2.
3.
4.
5.
Choices:
Addition Property
Definition of Congruent Segments
Given
Segment Addition Postulate
Substitution Property
4. Given: ∠2 ≅ ∠4
Prove: m∠1 = m∠3
Choices:
Congruent Supplements Theorem
Definition of Congruent Angles
Given
Linear Pair Theorem
5. Given: ∠WXY is a right angle, ∠1 ≅ ∠3
Prove: ∠1 and ∠2 are complementary
Statements
1. ∠WXY is a right angle
2. m∠WXY = 90°
3. m∠WXY = m∠2 + m∠3
4. m∠2 + m∠3 = 90°
5. ∠1 ≅ ∠3
6. m∠1 = m∠3
7. m∠2 + m∠1 = 90°
8. ∠1 and ∠2 are complementary
Reasons
1.
2.
3.
4.
5.
6.
7.
8.
Choices:
Angle Addition Postulate
Definition of Complementary Angles
Definition of Congruent Angles
Definition of Right Angle
Given
Substitution Property
6. Given: ∠1 and ∠2 are supplementary, ∠1 ≅ ∠2
Prove: ∠1 and ∠2 are right angles
Choices:
Definition of Congruent Angles
Definition of Right Angle
Definition of Supplementary Angles
Division Property
Given
Simplify
Substitution Property
9. Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3
Choices:
Given
Transitive Property
Vertical Angles Theorem
10. Given:
Prove:
⃗⃗⃗⃗⃗⃗
𝐵𝐷 bisects ∠ABC
⃗⃗⃗⃗⃗
𝐵𝐺 bisects ∠FBH
Choices:
Definition of Angle Bisector
Given
Transitive Property
Vertical Angles Theorem
11. Given: ∠3 is a right angle
Prove: ∠4 is a right angle
Choices:
Definition of Right Angle
Definition of Supplementary Angles
Given
Linear Pair Theorem
Substitution Property
Subtraction Property
12. Given: ∠2 ≅ ∠4
Prove: ∠1 ≅ ∠3
Statements
1. ∠2 ≅ ∠4
2. ∠1 ≅ ∠2, ∠3 ≅ ∠4
3. ∠1 ≅ ∠4
4. ∠1 ≅ ∠3
Reasons
1.
2.
3.
4.
Choices:
Given
Transitive Property
Vertical Angles Theorem