Question
Prove the following theorem: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
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)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
a. Response area
b. Response area
c. Response area
d. Response area
e. Response area
Statements
Reasons
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)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
a. Response area
b. Response area
c. Response area
d. Response area
e. Response area
Statements
Reasons
Answers
GPT-4o mini
To prove that quadrilateral ABCD is a parallelogram given that ∠A ≅ ∠C and ∠B ≅ ∠D, we can follow these steps:
```plaintext
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.
```plaintext
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.