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

Explanation:

Step1: Identify congruent alternate angles

$\angle 1 \cong \angle 3$ (Alternate Interior Angles Theorem, since $\overline{AB} \parallel \overline{DC}$ and $\overline{AC}$ is a transversal)

Step2: Identify second congruent angles

$\angle 2 \cong \angle 4$ (Alternate Interior Angles Theorem, since $\overline{AD} \parallel \overline{BC}$ and $\overline{AC}$ is a transversal)

Step3: Apply reflexive property to side

$\overline{AC} \cong \overline{AC}$ (Reflexive Property of Congruence)

Step4: Prove triangle congruence

$\triangle ABC \cong \triangle CDA$ (ASA Congruence Postulate, using $\angle 1 \cong \angle 3$, $\overline{AC} \cong \overline{AC}$, $\angle 4 \cong \angle 2$)

Step5: Corresponding parts of congruent triangles

$\overline{AD} \cong \overline{BC}$ (CPCTC: Corresponding Parts of Congruent Triangles are Congruent)

Answer:

Statement	Reason
2. $\overline{AB} \parallel \overline{DC}$	Definition of a Parallelogram
3. $\overline{AD} \parallel \overline{BC}$	Definition of a Parallelogram
4. $\angle 1 \cong \angle 3$	Alternate Interior Angles Theorem
5. $\angle 2 \cong \angle 4$	Alternate Interior Angles Theorem
6. $\overline{AC} \cong \overline{AC}$	Reflexive Property
7. $\triangle ABC \cong \triangle CDA$	ASA Congruence Postulate

| 8.

$$\begin{aligned} \overline{AB} \cong \overline{CD} \\ \overline{AD} \cong \overline{BC} \end{aligned}$$

| CPCTC (Corresponding Parts of Congruent Triangles are Congruent) |