quadrilateral abcd is a parallelogram. complete the statements to prove that $ \overline { a b } \cong \overline { c d } $ and $ \overline { b c } \cong \overline { a d } $.

| statement | reason |
| --- | --- |
| quadrilateral abcd is a parallelogram. | given |
| $ \overline { a b } \parallel \overline { c d } $ and $ \overline { b c } \parallel \overline { a d } $ | definition of a parallelogram |
| draw $ \overline { a c } $. $ \overline { a c } $ is a transversal that intersects two pairs of parallel lines, $ \overline { a b } $ and $ \overline { c d } $ and $ \overline { a d } $ and $ \overline { b c } $. | drawing a line segment |
| $ \angle c a b \cong \angle a c d $ and $ \angle b c a \cong \angle c a d $ | |
| $ m \angle c a b = m \angle a c d $ and $ m \angle b c a = m \angle c a d $ | congruent angles have equal measures. |
| $ \overline { a c } = \overline { a c } $ | |
| $ \delta a b c \cong \delta c d a $ | |
| $ \overline { a b } \cong \overline { c d } $ and $ \overline { b c } \cong \overline { a d } $ | corresponding sides of congruent triangles are congruent. |

Question

quadrilateral abcd is a parallelogram. complete the statements to prove that $ \overline { a b } \cong \overline { c d } $ and $ \overline { b c } \cong \overline { a d } $.

| statement | reason |
| --- | --- |
| quadrilateral abcd is a parallelogram. | given |
| $ \overline { a b } \parallel \overline { c d } $ and $ \overline { b c } \parallel \overline { a d } $ | definition of a parallelogram |
| draw $ \overline { a c } $. $ \overline { a c } $ is a transversal that intersects two pairs of parallel lines, $ \overline { a b } $ and $ \overline { c d } $ and $ \overline { a d } $ and $ \overline { b c } $. | drawing a line segment |
| $ \angle c a b \cong \angle a c d $ and $ \angle b c a \cong \angle c a d $ |  |
| $ m \angle c a b = m \angle a c d $ and $ m \angle b c a = m \angle c a d $ | congruent angles have equal measures. |
| $ \overline { a c } = \overline { a c } $ |  |
| $ \delta a b c \cong \delta c d a $ |  |
| $ \overline { a b } \cong \overline { c d } $ and $ \overline { b c } \cong \overline { a d } $ | corresponding sides of congruent triangles are congruent. |

Accepted Answer

# Explanation:
## Step1: Recall the Alternate Interior Angles Theorem
When a transversal intersects two parallel lines, alternate interior angles are congruent. Here, since $\overline{AB}\parallel\overline{CD}$ and $\overline{AC}$ is a transversal, $\angle CAB$ and $\angle ACD$ are alternate interior angles. Also, since $\overline{BC}\parallel\overline{AD}$ and $\overline{AC}$ is a transversal, $\angle BCA$ and $\angle CAD$ are alternate interior angles.
## Step2: Use the ASA (Angle - Side - Angle) Congruence Criterion
We know that $\angle CAB\cong\angle ACD$, $\overline{AC}=\overline{AC}$ (reflexive property of equality), and $\angle BCA\cong\angle CAD$. So, by the ASA congruence criterion, $\triangle ABC\cong\triangle CDA$.

# Answer:
The reason for $\angle CAB\cong\angle ACD$ and $\angle BCA\cong\angle CAD$ is the Alternate Interior Angles Theorem. The reason for $\triangle ABC\cong\triangle CDA$ is the ASA (Angle - Side - Angle) congruence criterion.

statement	reason
\\( \overline { a b } \parallel \overline { c d } \\) and \\( \overline { b c } \parallel \overline { a d } \\)	definition of a parallelogram
draw \\( \overline { a c } \\). \\( \overline { a c } \\) is a transversal that intersects two pairs of parallel lines, \\( \overline { a b } \\) and \\( \overline { c d } \\) and \\( \overline { a d } \\) and \\( \overline { b c } \\).	drawing a line segment
\\( \angle c a b \cong \angle a c d \\) and \\( \angle b c a \cong \angle c a d \\)
\\( m \angle c a b = m \angle a c d \\) and \\( m \angle b c a = m \angle c a d \\)	congruent angles have equal measures.
\\( \overline { a c } = \overline { a c } \\)
\\( \delta a b c \cong \delta c d a \\)
\\( \overline { a b } \cong \overline { c d } \\) and \\( \overline { b c } \cong \overline { a d } \\)	corresponding sides of congruent triangles are congruent.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the Alternate Interior Angles Theorem

Step2: Use the ASA (Angle - Side - Angle) Congruence Criterion

Answer: