Curriculum 2.0 Geometry: Unit 1 – Topic 4, SLT 41
Name:
Proving Parallelogram Properties Sample Answers
Date:
Period:
Proof 1
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
Complete the diagram and develop an appropriate “Given” and “Prove” for this case.
Given: 𝑄𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
̅̅̅̅ ≅ 𝐶𝐷
̅̅̅̅; 𝐴𝐷
̅̅̅̅ ≅ 𝐶𝐵
̅̅̅̅
Prove: 𝐴𝐵
B
A
D
C
Use triangle congruence criteria to demonstrate why opposite sides of a parallelogram are congruent.
Proof
Statement
Reason
1. 𝑄𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
1. Given
̅̅̅̅
2. ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐶𝐷; ̅̅̅̅
𝐴𝐷 ∥ 𝐶𝐵
2. Definition of a Parallelogram
3. ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐷𝐵
3. Alternate Interior Angles
4. ̅̅̅̅
𝐵𝐷 ≅ ̅̅̅̅
𝐵𝐷
4. Reflexive Property
5. ∠𝐶𝐵𝐷 ≅ ∠𝐴𝐷𝐵
5. Alternate Interior Angles
6. ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐷𝐵
6. ASA
̅̅̅̅
7. ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐶𝐷; ̅̅̅̅
𝐴𝐷 ≅ 𝐶𝐵
7. CPCTC
Page 1 of 4
Curriculum 2.0 Geometry: Unit 1 – Topic 4, SLT 41
Name:
Proving Parallelogram Properties Sample Answers
Date:
Period:
Proof 2
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
Complete the diagram and develop an appropriate “Given” and “Prove” for this case.
Given: 𝑄𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
Prove: ∠𝐵𝐴𝐷 ≅ ∠𝐷𝐶𝐵; 𝑚∠𝐴𝐷𝐶 = 𝑚∠𝐴𝐵𝐶
B
A
D
C
Use triangle congruence criteria to demonstrate why opposite angles of a parallelogram are congruent.
Proof
Statement
Reason
1. 𝑄𝑢𝑎𝑑𝑟𝑖𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝐴𝐵𝐶𝐷 𝑖𝑠 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
1. Given
̅̅̅̅
2. ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐶𝐷; ̅̅̅̅
𝐴𝐷 ∥ 𝐶𝐵
2. Definition of a Parallelogram
3. ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐷𝐵; ∠𝐶𝐵𝐷 ≅ ∠𝐴𝐷𝐵
3. Alternate Interior Angles
̅̅̅̅ ≅ 𝐵𝐷
̅̅̅̅
4. 𝐵𝐷
4. Reflexive Property of Equality
5. ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐷𝐵
5. ASA
6. ∠𝐵𝐴𝐷 ≅ ∠𝐷𝐶𝐵
6. CPCTC
7. 𝑚∠𝐴𝐷𝐵 + 𝑚∠𝐶𝐷𝐵 = 𝑚∠𝐴𝐷𝐶;
7. Angle Addition Postulate
𝑚∠𝐴𝐵𝐷 + 𝑚∠𝐶𝐵𝐷 = 𝑚∠𝐴𝐵𝐶
8. 𝑚∠𝐶𝐵𝐷 + 𝑚∠𝐴𝐵𝐷 = 𝑚∠𝐴𝐷𝐶
8. Substitution from #3 into #7
9. 𝑚∠𝐴𝐷𝐶 = 𝑚∠𝐴𝐵𝐶
9. Substitution or Transitive Property
Page 2 of 4
Curriculum 2.0 Geometry: Unit 1 – Topic 4, SLT 41
Name:
Proving Parallelogram Properties Sample Answers
Date:
Period:
The converse of Proof 1
If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
Complete the diagram and develop an appropriate “Given” and “Prove” for this case.
̅̅̅̅
Given: Quadrilateral ABCD with ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐶𝐷; ̅̅̅̅
𝐴𝐷 ≅ 𝐶𝐵
Prove: Quadrilateral ABCD is a parallelogram
B
A
D
C
Use triangle congruence criteria to determine that quadrilateral ABCD is a parallelogram.
Proof
Statement
Reason
̅̅̅̅ ≅ 𝐶𝐷
̅̅̅̅; 𝐴𝐷
̅̅̅̅ ≅ 𝐶𝐵
̅̅̅̅
1. Quadrilateral ABCD with 𝐴𝐵
1. Given
̅̅̅̅ ≅ 𝐵𝐷
̅̅̅̅
2. 𝐵𝐷
2. Reflexive Property
3. ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐷𝐵
3. SSS
4. ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐷𝐵, ∠𝐴𝐷𝐵 ≅ ∠𝐶𝐵𝐷
4. CPCTC
̅̅̅̅ ∥ 𝐶𝐷
̅̅̅̅, 𝐴𝐷
̅̅̅̅ ∥ 𝐶𝐵
̅̅̅̅
5. 𝐴𝐵
5. Converse of Alternate Interior Angles
6. Quadrilateral ABCD is a parallelogram
6. Definition of a Parallelogram
Page 3 of 4
Curriculum 2.0 Geometry: Unit 1 – Topic 4, SLT 41
Name:
Proving Parallelogram Properties Sample Answers
Date:
Period:
Converse of Proof 2
If the opposite angles of a quadrilateral are equal, then the quadrilateral is a parallelogram.
A
Complete the diagram and develop an appropriate “Given” and “Prove” for this case. 1
Given: 𝑚∠𝐶𝐷𝐴 = 𝑚∠𝐴𝐵𝐶; 𝑚∠1 = 𝑚∠4
B
2
3
6
D
Prove: Quadrilateral ABCD is a parallelogram
5
4
C
Use triangle congruence criteria to demonstrate that quadrilateral ABCD is a parallelogram.
Proof
Statements
Reasons
1. mCDA  mABC
1. Given
mCDA  m5  m6
mABC  m2  m3
3. m5  m6  m2  m3
2. Angle Addition
2.
4.
m1  m2  m6  180o
m3  m4  m5  180
o
5. m1  m2  m6  m3  m4  m5
6. m1  m4
7. m4  m2  m6  m3  m4  m5
8. m2  m6  m3  m5
9. m5  m2  m2  m5
10. 2m5  2m2
3. Substitution or Transitive (statements from #2 and#1)
4. Sum of the measures of the angles of a triangle is 180o
5. Substitution or Transitive (both expressions in statement #4
equal 180o)
6. Given
7. Substitution (replacing m1 from statement #5 with m4
in statement #6)
8. Subtraction Property of Equality (subtract m4 from both
sides)
9. Subtraction Property of Equality (statement #3 minus
statement #8)
10. Addition Property of Equality (add m2 and m5 to
both sides)
11. m5  m2
11. Division Property of Equality (divide both sides by 2)
12. AB / /CD
12. Converse of Alternate Interior Angles
13. m5  m6  m3  m5
13. Substitution (replacing m2 in statement #8 with m5
from statement #11)
14. Subtraction Property of Equality (subtract m5 from both
sides)
14. m6  m3
15. AD / / BC
16. Quadrilateral ABCD is a parallelogram
15. Converse of Alternate Interior Angles
16. Definition of a parallelogram
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