Prove the following theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Question

Given: ∠A ≅ ∠C, ∠B ≅ ∠D

Prove: ABCD is a parallelogram.

Quadrilateral ABCD has interior angle A congruent to angle C, which measures y degrees, and angle D congruent to angle B, which measures x degrees.



(5 points)
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a. Response area

b. Response area

c. Response area

d. Response area

e. Response area

Statements
Reasons

Answer 1

To prove that quadrilateral ABCD is a parallelogram given that ∠A ≅ ∠C and ∠B ≅ ∠D, we can follow these steps:

Statements                                     | Reasons
-----------------------------------------------------------
1. ∠A ≅ ∠C and ∠B ≅ ∠D                         | Given
2. m∠A + m∠B + m∠C + m∠D = 360°               | Sum of interior angles of a quadrilateral
3. m∠A + m∠B + m∠A + m∠B = 360°               | Substitute ∠C with ∠A and ∠D with ∠B (from statement 1)
4. 2m∠A + 2m∠B = 360°                          | Combine like terms
5. m∠A + m∠B = 180°                            | Divide both sides by 2
6. ∠A + ∠B = 180° and ∠C + ∠D = 180°          | ∠C = ∠A and ∠D = ∠B (from statement 1)
7. ABCD is a parallelogram                      | If two pairs of opposite angles are supplementary, then the quadrilateral is a parallelogram

Matching Responses:

a. ∠A ≅ ∠C and ∠B ≅ ∠D
b. m∠A + m∠B + m∠C + m∠D = 360°
c. m∠A + m∠B + m∠A + m∠B = 360°
d. 2m∠A + 2m∠B = 360°
e. m∠A + m∠B = 180°
f. ∠A + ∠B = 180° and ∠C + ∠D = 180°
g. ABCD is a parallelogram

This proof shows that if both pairs of opposite angles of a quadrilateral are congruent, then the angles sum to 180 degrees in pairs, which establishes that the quadrilateral is a parallelogram.