Question

tyler has written an incorrect proof to show that quadrilateral abcd is a parallelogram. he knows segments ab and dc are congruent. he also knows angles abc and adc are congruent. find the mistake in his proof. segment ac is congruent to itself, so triangle abc is congruent to triangle adc by side - angle - side triangle congruence theorem. since the triangles are congruent, so are the corresponding parts, and so angle dac is congruent to acb. in quadrilateral abcd, ab is congruent to cd and ad is parallel to cb. since ad is parallel to cb, alternate interior angles dac and bca are congruent. since alternate interior angles are congruent, ab must be parallel to cd. quadrilateral abcd must be a parallelogram since both pairs of opposite sides are parallel.

Explanation:

Step1: Analyze triangle - congruence claim

The proof uses Side - Angle - Side (SAS) to claim $\triangle ABC\cong\triangle ADC$. But the given information only has $AB = DC$ and $\angle ABC=\angle ADC$ and $AC = AC$. For SAS, the angle must be the included angle between the two sides. Here, the angle is not the included angle, so the claim of $\triangle ABC\cong\triangle ADC$ by SAS is incorrect.

Answer:

The mistake is using Side - Angle - Side (SAS) triangle congruence theorem incorrectly as the given angle is not the included angle between the two given sides when trying to prove $\triangle ABC\cong\triangle ADC$.