Question

given: abcd is a parallelogram
prove: ∠abc≅∠cda
statement: reason:
abcd is a parallelogram

1. a. opposite angles of a parallelogram are congruent b. cpctc c. definition of parallelogram d. given

ba∥dc

1. a. definition of parallelogram (opposite sides are parallel) b. alternate interior angles c. opposite sides of a parallelogram are congruent d. reflexive property
2. alternate interior angles a. ∠adc≅∠dac b. ∠bac≅∠dca c. ∠cab≅∠dac d. ∠bca≅∠dca
3. opposite sides of a parallelogram are congruent a. ab≅ab b. ab≅dc c. ab≅ac d. ab≅ad

ac≅ac

1. a. reflexive property b. diagonals bisect each other c. transitive property d. symmetric property

∠abc≅∠cda

1. a. aas b. sas c. hl d. asa

∠abc≅∠cda

1. a. alternate interior angles b. vertical angles c. definition of parallelogram d. cpctc

Explanation:

Step1: Identify the reason for the given statement

The statement "ABCD is a parallelogram" is given in the problem - set up. So the reason is "Given".

Step2: Reason for $\overline{BA}\parallel\overline{DC}$

By the definition of a parallelogram, opposite sides are parallel. So $\overline{BA}\parallel\overline{DC}$ because of the definition of a parallelogram.

Step3: Find the alternate - interior angles

Since $\overline{BA}\parallel\overline{DC}$ and $\overline{AC}$ is a transversal, $\angle BAC\cong\angle DCA$ by the alternate - interior angles theorem.

Step4: Identify congruent sides of the parallelogram

Opposite sides of a parallelogram are congruent. So $\overline{AB}\cong\overline{DC}$.

Step5: Reason for $\overline{AC}\cong\overline{AC}$

The side $\overline{AC}$ is common to both $\triangle ABC$ and $\triangle CDA$. By the reflexive property, $\overline{AC}\cong\overline{AC}$.

Step6: Prove the triangles congruent

In $\triangle ABC$ and $\triangle CDA$, we have $\angle BAC\cong\angle DCA$, $\overline{AB}\cong\overline{DC}$, and $\overline{AC}\cong\overline{AC}$. So $\triangle ABC\cong\triangle CDA$ by the SAS (Side - Angle - Side) congruence criterion.

Step7: Prove the angles congruent

Since $\triangle ABC\cong\triangle CDA$, then $\angle ABC\cong\angle CDA$ by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Answer:

1. D. Given
2. A. Definition of parallelogram (Opposite sides are parallel)
3. B. $\angle BAC\cong\angle DCA$
4. B. $\overline{AB}\cong\overline{DC}$
5. A. Reflexive property
6. B. SAS
7. D. CPCTC