Question

use the given information to complete the proof of the following theorem.
if opposite sides of a quadrilateral are congruent, then it is a parallelogram.
by definition, a parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
use this definition in your proof.
given: $overline{ab} \cong \overline{cd}$
$overline{ad} \cong \overline{bc}$
prove: $abcd$ is a parallelogram
statement reason
1 $overline{ab} \cong \overline{cd}$ given
2 $overline{ad} \cong \overline{bc}$ given
3 $\overline{\square} \cong \overline{\square}$ reason?
4 $\triangle abd \cong \triangle \square$ reason?
5 $\angle 1 \cong \angle \square$ reason?
6 $\angle 2 \cong \angle \square$ reason?
7 $overline{ab} \parallel \overline{\square}$ if alt. int. $\angle$s $\cong$, then lines $\parallel$
8 $overline{ad} \parallel \overline{\square}$ if alt. int. $\angle$s $\cong$, then lines $\parallel$
9 $abcd$ is a parallelogram reason?

Explanation:

Step 1: Identify the common side

In triangles \( \triangle ABD \) and \( \triangle CDB \), the side \( \overline{BD} \) is common to both triangles. So, \( \overline{BD} \cong \overline{BD} \) by the Reflexive Property of Congruence.

Step 2: Prove triangle congruence

We have \( \overline{AB} \cong \overline{CD} \) (given), \( \overline{AD} \cong \overline{BC} \) (given), and \( \overline{BD} \cong \overline{BD} \) (reflexive). By the SSS (Side - Side - Side) Congruence Postulate, \( \triangle ABD \cong \triangle CDB \).

Step 3: Corresponding angles congruence

Since \( \triangle ABD \cong \triangle CDB \), their corresponding angles are congruent. So, \( \angle 1\cong\angle 3 \) and \( \angle 2\cong\angle 4 \) (Corresponding Parts of Congruent Triangles are Congruent - CPCTC).

Step 4: Prove sides are parallel

• For \( \overline{AB} \parallel \overline{CD} \): \( \angle 1 \) and \( \angle 3 \) are alternate interior angles. Since \( \angle 1\cong\angle 3 \), by the Alternate Interior Angles Theorem, \( \overline{AB} \parallel \overline{CD} \).
• For \( \overline{AD} \parallel \overline{BC} \): \( \angle 2 \) and \( \angle 4 \) are alternate interior angles. Since \( \angle 2\cong\angle 4 \), by the Alternate Interior Angles Theorem, \( \overline{AD} \parallel \overline{BC} \).

Step 5: Conclude it's a parallelogram

Since both pairs of opposite sides (\( \overline{AB} \parallel \overline{CD} \) and \( \overline{AD} \parallel \overline{BC} \)) of quadrilateral \( ABCD \) are parallel, by the definition of a parallelogram, \( ABCD \) is a parallelogram.

Now, filling in the table:

Statement 3:

\( \overline{BD} \cong \overline{BD} \)
Reason: Reflexive Property of Congruence

Statement 4:

\( \triangle ABD \cong \triangle CDB \)
Reason: SSS Congruence Postulate

Statement 5:

\( \angle 1\cong\angle 3 \)
Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Statement 6:

\( \angle 2\cong\angle 4 \)
Reason: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Statement 7:

\( \overline{AB} \parallel \overline{CD} \)
Reason: If alternate interior angles are congruent, then lines are parallel (Alternate Interior Angles Theorem)

Statement 8:

\( \overline{AD} \parallel \overline{BC} \)
Reason: If alternate interior angles are congruent, then lines are parallel (Alternate Interior Angles Theorem)

Statement 9:

\( ABCD \) is a parallelogram
Reason: Definition of a parallelogram (both pairs of opposite sides are parallel)

Answer:

1. \( \boldsymbol{\overline{BD} \cong \overline{BD}} \); Reason: Reflexive Property of Congruence
2. \( \boldsymbol{\triangle ABD \cong \triangle CDB} \); Reason: SSS Congruence Postulate
3. \( \boldsymbol{\angle 1\cong\angle 3} \); Reason: CPCTC
4. \( \boldsymbol{\angle 2\cong\angle 4} \); Reason: CPCTC
5. \( \boldsymbol{\overline{AB} \parallel \overline{CD}} \); Reason: Alternate Interior Angles Theorem
6. \( \boldsymbol{\overline{AD} \parallel \overline{BC}} \); Reason: Alternate Interior Angles Theorem
7. \( \boldsymbol{ABCD} \) is a parallelogram; Reason: Definition of a parallelogram