Question

for items 11 - 13, refer to the diagram. if (overline{ab}paralleloverline{cd}), is (\triangle acdcong\triangle cab)? explain. a. yes, the triangles are congruent by angle - angle - side. b. yes, the triangles are congruent by angle - side - angle. c. yes, the triangles are congruent by side - angle - side. d. no, the triangles are not congruent.

Explanation:

Step1: Identify equal - side and angles

Since $\overline{AB}\parallel\overline{CD}$, we have $\angle BAC=\angle DCA$ (alternate - interior angles). Also, $\overline{AC}$ is common to both $\triangle ACD$ and $\triangle CAB$. And we can assume from the markings that $\overline{AB}=\overline{CD}$.

Step2: Apply congruence criterion

We have two angles and the included side equal. $\angle BAC=\angle DCA$, $\overline{AC}=\overline{CA}$, and $\angle ACD=\angle CAB$ (by the property of parallel lines and the common side). So, $\triangle ACD\cong\triangle CAB$ by the Angle - Side - Angle (ASA) congruence criterion.

Answer:

A. Yes, the triangles are congruent by Angle - Side - Angle