Name: _________________________________
Date: __________________ Period: ________
Geometric Proofs
∠𝐴 ≅ ∠𝐴
𝐼𝑓 ∠𝐴 ≅ ∠𝐵, . 𝑡ℎ𝑒𝑛 ∠𝐵 ≅ ∠𝐴.
𝐼𝑓 ∠𝐴 ≅ ∠𝐵 𝑎𝑛𝑑 ∠𝐵 ≅ ∠𝐶, 𝑡ℎ𝑒𝑛 ∠𝐴 ≅ ∠𝐶.
𝐼𝑓 ∠𝐴 ≅ ∠𝐵, . 𝑡ℎ𝑒𝑛 𝑚∠𝐵 = 𝑚∠𝐴.
Example 1:
Given: ∠A and ∠B are complementary, ∠A ≅ ∠C
Prove: ∠C and ∠B are complementary
Statements:
1. ∠A and ∠B are complementary
2. m∠A + m∠B = 90°
3. ∠A ≅ ∠C
4. m∠A = m∠C
5. m∠C + m∠B = 90°
6. ∠C and ∠B are complementary
Reasons:
1.
2.
3.
4.
5.
6.
Example 2:
Given: B is the midpoint of ̅̅̅̅
𝐴𝐶 , ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐸𝐹
Prove: ∠C and ∠B are complementary
Statements:
̅̅̅̅
1. B is the midpoint of 𝐴𝐶
̅̅̅̅
̅̅̅̅
2. 𝐵𝐶 ≅ 𝐴𝐵
3. ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐸𝐹
̅̅̅̅ ≅ ̅̅̅̅
4. 𝐵𝐶
𝐸𝐹
Reasons:
1.
2.
3.
4.
Choices:
Definition of Complementary
Definition of Congruence
Given
Substitution Property of Equality
Choices:
Definition of Midpoint
Given
Transitive Property of Congruence
Example 3:
Fill in the blanks to complete the two-column proof.
Given: ∠1 and ∠2 form a linear pair.
Prove: ∠1 and ∠2 are supplementary
Choices:
Definition of Supplementary
m∠1 + m∠2 = 180°
m∠1 + m∠2 = m∠ABC
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Example 4:
Fill in the blanks to complete the two-column proof.
Given: ∠1 and ∠2 are supplementary, ∠2 and ∠3 are supplementary
Prove: ∠1 ≅ ∠3
Choices:
Subtraction Property of Equality
∠1  ∠3
m∠1 + m∠2 = m∠2 + m∠3
∠1 and ∠2 are supplementary
∠2 and ∠3 are supplementary
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Example 5:
Complete the two-column proof.
Given: ∠1 and ∠2 are right angles
Prove: ∠1 ≅ ∠2
Choices:
Definition of Congruent Angle
Definition of Right Angles
Given
Substitution Property of Equality
Practice:
1. Given: m∠A = 60°, m∠B = 2m∠A
Prove: ∠A and ∠B are supplementary
Statements:
1. m∠A = 60°, m∠B = 2m∠A
2. m∠B = 2(60°)
3. m∠B = 120°
4. m∠A + m∠B = 60° + 120°
5. m∠A + m∠B = 180°
6. ∠A and ∠B are supplementary
Reasons:
1.
2.
3.
4.
5.
6.
Choices:
Definition of Supplementary
Given
Simplify
Substitution Property of Equality
2. Given: ∠2 ≅ ∠3
Prove: ∠1 and ∠3 are supplementary
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Choices:
∠1 and ∠3 are supplementary
∠1 and ∠2 are supplementary
Definition of Congruent Angles
Substitution Property of Equality
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3. Given: X is the midpoint of ̅̅̅̅
𝐴𝑌, Y is the midpoint of ̅̅̅̅
𝑋𝐵
̅̅̅̅
̅̅̅̅
Prove: 𝐴𝑋 ≅ 𝑌𝐵
Statements:
1. X is the midpoint of ̅̅̅̅
𝐴𝑌,
Y is the midpoint of ̅̅̅̅
𝑋𝐵
2. ̅̅̅̅
𝐴𝑋 ≅ ̅̅̅̅
𝑋𝑌
̅̅̅̅ ≅ ̅̅̅̅
𝑋𝑌
𝑌𝐵
3. ̅̅̅̅
𝐴𝑋 ≅ ̅̅̅̅
𝑌𝐵
Reasons:
1.
2.
Choices:
Definition of Midpoint
Given
Transitive Property of Equality
3.
4. Given: ⃗⃗⃗⃗⃗
𝐵𝑋 bisects ∠ABC, m∠XBC = 45°
Prove: ∠ABC is a right angle
Statements:
1. ⃗⃗⃗⃗⃗
𝐵𝑋 bisects ∠ABC
2. ∠ABX ≅ ∠XBC
3. m∠ABX = m∠XBC
4. m∠XBC = 45°
5. m∠ABX = 45°
6. m∠ABX + m∠XBC = m∠ABC
7. 45° + 45° = m∠ABC
8. 90° = m∠ABC
9. ∠ABC is a right angle
Reasons:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Choices:
Angle Addition Postulate
Definition of Angle Bisector
Definition of Congruent Angles
Definition of Right Angle
Given
Simplify
Substitution Property of Equality
5. Given: ∠1 and ∠2 are supplementary, ∠3 and ∠4 are supplementary, ∠2 ≅ ∠3
Prove: ∠1 ≅ ∠4
Choices:
Definition of Congruent Angles
Substitution Property of Equality
m∠1 = m∠4
m∠1 + m∠2 = 180°
m∠3 + m∠4 = 180°
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6. Given: ∠BAC is a right angle, ∠2 ≅ ∠3
Prove: ∠1 and ∠3 are complementary
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Choices:
∠1 and ∠3 are complementary angles
m∠1 + m∠2 = m∠BAC
m∠2 = m∠3
Definition of Right Angles
Substitution Property of Equality
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7. Given: ∠1 and ∠3 are complementary, ∠2 and ∠4 are complementary, ∠3 ≅ ∠4
Prove: ∠1 ≅ ∠2
Statements:
1. ∠1 and ∠3 are complementary
∠2 and ∠4 are complementary
2. m∠1 + m∠3 = 90°
m∠2 + m∠4 = 90°
3. m∠1 + m∠3 = m∠2 + m∠4
4. ∠3 ≅ ∠4
5. m∠3 = m∠4
6. m∠1 + m∠4 = m∠2 + m∠4
7. m∠1 = m∠2
8. ∠1 ≅ ∠2
Reasons:
1.
2.
3.
4.
5.
6.
7.
8.
Choices:
Definition of Complementary Angles
Definition of Congruence
Given
Substitution Property of Equality
Subtraction Property of Equality