2-6 Geometric Proof Homework
1. In a two column proof, you list the statements in the left column and the reasons in the right
column.
2. A theorem is a statement you can prove.
3. Write a justification for each step, given that m∠A = 60° and m∠B = 2m∠A.
1.
2.
3.
4.
5.
6.
Statements
m∠A = 60°, m∠B = 2m∠A
m∠B = 2(60°)
m∠B = 120°
m∠A + m∠B = 60° + 120°
m∠A + m∠B = 180°
∠A and ∠B are supplementary
Reasons
1.
2.
3.
4.
5.
6.
Given
Substitution
Simplify
Addition Property of Equality
Simplify
Definition of Supplementary Angles
4. Fill in the blanks to complete the two column proof.
Given: ∠2 ≅ ∠3
Prove: ∠1 and ∠3 are supplementary.
1.
2.
3.
4.
5.
6.
Statements
∠2 ≅ ∠3
m∠2 = m∠3
b. ∠1 and ∠2 are supplementary angles
m∠1 + m∠2 =180°
m∠1 + m∠3 =180°
d. ∠1 and ∠3 are supplementary
1.
2.
3.
4.
5.
6.
Reasons
Given
a. Definition of Congruent Angles
Linear Pair Theorem
Definition of Supplementary Angles
c. Substitution
Definition of Supplementary Angles
5. Use the given plan to write a two-column proof.
Given: X is the midpoint of ̅̅̅̅
𝐴𝑌, and Y is the midpoint of ̅̅̅̅
𝑋𝐵.
Prove: ̅̅̅̅
𝐴𝑋≅̅̅̅̅
𝑌𝐵
̅̅̅̅≅𝑋𝑌
̅̅̅̅, and 𝑋𝑌
̅̅̅̅≅𝑌𝐵
̅̅̅̅ .
Plan: By the definition of midpoint, 𝐴𝑋
̅̅̅̅≅𝑌𝐵
̅̅̅̅.
Use the Transitive Property to conclude that 𝐴𝑋
Statements
1. X is the midpoint ̅̅̅̅
𝐴𝑌
Y is the midpoint of ̅̅̅̅
𝑋𝐵
̅̅̅̅
2. ̅̅̅̅
𝐴𝑋 ≅ 𝑋𝑌
̅̅̅̅
̅̅̅̅
𝑋𝑌 ≅ 𝑌𝐵
̅̅̅̅
3. 𝐴𝑋 ≅ ̅̅̅̅
𝑌𝐵
Reasons
1. Given
2. Definition of midpoint
3. Transitive Property of Congruence
6. Write a justification for each step , given that ⃗⃗⃗⃗⃗
𝐵𝑋 bisects ∠ABC and m∠XBC = 45°.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Statements
BX bisects ∠ABC
∠ABX ≅ ∠XBC
m∠ABX ≅ m∠XBC
m∠XBC = 45°
m∠ABX = 45°
m∠ABX + m∠XBC = m∠ABC
45° + 45° = m∠ABC
90° = m∠ABC
∠ABC is a right angle
Reasons
1.
2.
3.
4.
5.
6.
7.
8.
9.
Given
Definition of Angle Bisector
Definition of Congruent Angles
Given
Substitution
Angle Addition Postulate
Substitution
Simplify
Definition of Right Angles
Fill in the blanks to complete each two column proof.
7. Given: ∠1 and ∠2 are supplementary, and ∠3 and ∠4 are supplementary. ∠2 ≅ ∠3
Prove: ∠1 ≅ ∠4
1.
2.
3.
4.
5.
6.
7.
Statements
∠1 and ∠2 are supplementary
∠3 and ∠4 are supplementary
a. m∠1 + m∠2 =180°
m∠1 + m∠3 =180°
m∠1 + m∠2 = m∠3 + m∠4
∠2 ≅ ∠3
m∠2 = m∠3
c. m∠1 = m∠4
∠1 ≅ ∠4
Reasons
1. Given
2. Definition of Supplementary Angles
3.
4.
5.
6.
7.
Substitution
Given
Definition of Congruent Angles
Subtraction Property of Equality
Definition of Congruent Angles
8. Given: ∠BAC is a right angle. ∠2 ≅ ∠3
Prove: ∠1 and ∠3 are complementary.
1.
2.
3.
4.
5.
6.
7.
8.
Statements
∠BAC is a right angle
m∠BAC = 90°
b. m∠1 + m∠2 = m∠BAC
m∠1 + m∠2 =90°
∠2 ≅ ∠3
c. m∠2 ≅ m∠3
m∠1 + m∠3 =90°
e. ∠1 and ∠3 are complementary
Reasons
1.
2.
3.
4.
5.
6.
7.
8.
Given
a. Definition of right angles
Angle Addition Postulate
Substitution
Given
Definition of Congruent Angles
d. Substitution
Definition of Complementary Angles
9. Use the given plan to write a two-column proof.
Given: ̅̅̅̅
𝑩𝑬 ≅ ̅̅̅̅
𝑪𝑬, ̅̅̅̅
𝑫𝑬 ≅ ̅̅̅̅
𝑨𝑬
̅̅̅̅
̅̅̅̅ ≅ 𝑪𝑫
Prove: 𝑨𝑩
Plan: Use the definition of congruent segments to write the given information in terms of lengths.
̅̅̅̅ ≅ 𝐶𝐷
̅̅̅̅.
Then use the Segment Addition Postulate to show that AB = CD and thus 𝐴𝐵
Statements
1. BE ≅ CE
DE ≅ AE
2. BE = CE
DE = AE
3. AE + BE = AB
CE + DE = CD
4. DE + CE = AB
5. AB = CD
6. AB ≅ CD
Reasons
1. Given
2. Definition of congruent segments
3. Segment Addition Postulate
4. Substitution
5. Substitution
6. Definition of congruent segments
10. Given: ∠1 and ∠3 are complementary, and ∠2 and ∠4 are complementary. ∠3 ≅ ∠4
Prove: ∠1 ≅ ∠2
Plan: Since ∠1 and ∠3 are complementary and ∠2 and ∠4 are complementary, both pairs of
angle measures add to 90°. Use substitution to show that the sums of both pairs are equal.
Since ∠3 ≅ ∠ 4, their measures are equal. Use the Subtraction Property of Equality and the
definition of congruent angles to conclude that ∠1 ≅ ∠2.
1.
2.
3.
4.
5.
6.
Statements
∠1 and ∠3 are complementary
∠2 and ∠4 are complementary
∠3 ≅ ∠4
m∠1 + m∠3 =90°
m∠2 + m∠4 =90°
m∠1 + m∠3 = m∠2 + m∠4
m∠3 = m∠4
m∠1 = m∠2
∠1 ≅ ∠2
11. m∠1 = 180° - 48° = 132°
12. m∠2 = 90° - 63° = 27°
13. m∠3 = 90° - 31° = 59°
Reasons
1. Given
2. Definition of Complementary Angles
3.
4.
5.
6.
Substitution
Definition of Congruent Angles
Subtraction Property of Equality
Definition of Congruent Angles
16.
17.
18.
19.
20.
Sometimes
Sometimes
Sometimes
Never
4𝑛 + 5 + 8𝑛 − 5 = 180°
12𝑛 = 180°
𝑛 = 15°
21. 9𝑥 − 6 = 8.5𝑥 + 2
. 5𝑥 − 6 = 2
. 5𝑥 = 8
𝑥 = 16
22. 4𝑧 + 3𝑧 + 6 = 90°
7𝑧 + 6 = 90°
7𝑧 = 84°
𝑧 = 12°