Question

use the given information to complete the proof of the following theorem.
if a quadrilateral is a parallelogram, then its opposite sides are congruent.
by definition, a parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
use this definition in your proof.
given: (abcd) is a parallelogram
prove: (overline{ad} cong overline{bc})

statement	reason
2 (overline{ab} parallel square)	definition of a parallelogram
3 (overline{ad} parallel square)	definition of a parallelogram
4 (angle 1 cong angle square)	reason?
5 (angle 2 cong angle square)	reason?
6 (square cong square)	reflexive property
7 (\triangle abc cong \triangle square)	reason?
8 (overline{ab} cong overline{cd}) (overline{ad} cong overline{bc})	reason?

Explanation:

Step 1: Analyze \( \overline{AB} \parallel \)

In a parallelogram \(ABCD\), by the definition of a parallelogram (both pairs of opposite sides are parallel), \( \overline{AB} \parallel \overline{CD} \). So the missing side here is \( \overline{CD} \).

Step 2: Analyze \( \overline{AD} \parallel \)

Again, using the definition of a parallelogram, \( \overline{AD} \parallel \overline{BC} \). So the missing side here is \( \overline{BC} \).

Step 3: Analyze \( \angle 1 \cong \angle \)

Since \( \overline{AB} \parallel \overline{CD} \) and \( \overline{AC} \) is a transversal, the alternate interior angles \( \angle 1 \) and \( \angle 3 \) are congruent (Alternate Interior Angles Theorem). So the missing angle here is \( 3 \) (i.e., \( \angle 3 \)).

Step 4: Analyze \( \angle 2 \cong \angle \)

Since \( \overline{AD} \parallel \overline{BC} \) and \( \overline{AC} \) is a transversal, the alternate interior angles \( \angle 2 \) and \( \angle 4 \) are congruent (Alternate Interior Angles Theorem). So the missing angle here is \( 4 \) (i.e., \( \angle 4 \)).

Step 5: Analyze \( \overline{\quad} \cong \overline{\quad} \) (Reflexive Property)

The reflexive property states that a segment is congruent to itself. So \( \overline{AC} \cong \overline{AC} \).

Step 6: Analyze \( \triangle ABC \cong \triangle \)

We have \( \angle 1 \cong \angle 3 \), \( \overline{AC} \cong \overline{AC} \), and \( \angle 2 \cong \angle 4 \). By the Angle - Side - Angle (ASA) Congruence Postulate, \( \triangle ABC \cong \triangle CDA \).

Step 7: Analyze \( \overline{AD} \cong \overline{BC} \)

Since \( \triangle ABC \cong \triangle CDA \), their corresponding sides are congruent. So \( \overline{AD} \cong \overline{BC} \) (Corresponding Parts of Congruent Triangles are Congruent - CPCTC).

Answer:

1. \( \overline{AB} \parallel \boldsymbol{\overline{CD}} \)
2. \( \overline{AD} \parallel \boldsymbol{\overline{BC}} \)
3. \( \angle 1 \cong \boldsymbol{\angle 3} \) (Alternate Interior Angles Theorem)
4. \( \angle 2 \cong \boldsymbol{\angle 4} \) (Alternate Interior Angles Theorem)
5. \( \boldsymbol{\overline{AC}} \cong \boldsymbol{\overline{AC}} \)
6. \( \triangle ABC \cong \boldsymbol{\triangle CDA} \) (ASA Congruence Postulate)
7. \( \overline{AD} \cong \overline{BC} \) (CPCTC)