Question

fall 2025 geometry b robichaud proving a quadrilateral is a parallelogram proving the single opposite side pair theorem given $overline{ad}congoverline{bc}$ and $overline{ad}paralleloverline{bc}$ prove $abcd$ is a parallelogram. statements reasons 1 $overline{ad}congoverline{bc}$ 1. given 2 $overline{ad}paralleloverline{bc}$ 2. given 3 $angle dac$ and $angle bca$ are alt. interior angles 3. def of alt. interior angles 4 $angle daccongangle bca$ 4. alternate interior angles theorem 5 $overline{ac}congoverline{ac}$ 5. reflexive property 6 $\triangle daccong\triangle bca$ 6. sas 7 $overline{ab}congoverline{cd}$ 7 cpctc

Explanation:

Step1: Recall given information

Given $\overline{AD}\cong\overline{BC}$ and $\overline{AD}\parallel\overline{BC}$.

Step2: Identify alternate - interior angles

Since $\overline{AD}\parallel\overline{BC}$, $\angle DAC$ and $\angle BCA$ are alternate - interior angles (by definition of alternate - interior angles).

Step3: Apply alternate - interior angles theorem

By the alternate - interior angles theorem, $\angle DAC\cong\angle BCA$.

Step4: Use reflexive property

$\overline{AC}\cong\overline{AC}$ by the reflexive property of congruence.

Step5: Prove triangle congruence

In $\triangle DAC$ and $\triangle BCA$, we have $\overline{AD}\cong\overline{BC}$, $\angle DAC\cong\angle BCA$, and $\overline{AC}\cong\overline{AC}$, so $\triangle DAC\cong\triangle BCA$ by the Side - Angle - Side (SAS) congruence criterion.

Step6: Use CPCTC

Since $\triangle DAC\cong\triangle BCA$, by Corresponding Parts of Congruent Triangles are Congruent (CPCTC), $\overline{AB}\cong\overline{CD}$.

Step7: Recall parallelogram definition

A quadrilateral with two pairs of opposite sides congruent is a parallelogram. We have $\overline{AD}\cong\overline{BC}$ and $\overline{AB}\cong\overline{CD}$, so $ABCD$ is a parallelogram.

Answer:

$ABCD$ is a parallelogram.