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

Explanation:

Step1: Identify common side

In $\triangle ADC$ and $\triangle CBA$, $AC = AC$ by the reflexive - property.

Step2: Apply SAS congruence

We have $AD\cong BC$ (given), $\angle DAC\cong\angle BCA$ (alternate - interior angles theorem) and $AC = AC$ (reflexive property). So, $\triangle ADC\cong\triangle CBA$ by SAS (Side - Angle - Side) congruence criterion.

Step3: Use CPCTC

By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $AB\cong DC$.

Step4: Apply parallelogram side theorem

Since $AD\cong BC$ and $AB\cong DC$, by the parallelogram side theorem (if both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram), $ABCD$ is a parallelogram.

Answer:

1. $\triangle ADC\cong\triangle CBA$, SAS; 6. $AB\cong DC$, CPCTC; 7. $ABCD$ is a parallelogram, parallelogram side theorem.