Question

given ad||bc and ab||dc, name as many pairs of congruent triangles as you can. select all that apply. a. △ade and △cbe b. △ade and △cde c. △abc and △cda d. △abe and △cde e. △abe and △cbe f. △abd and △cdb

Explanation:

Step1: Identify the parallelogram

Since $\overline{AD}\parallel\overline{BC}$ and $\overline{AB}\parallel\overline{DC}$, $ABCD$ is a parallelogram. In a parallelogram, $AB = DC$, $AD=BC$, and the diagonals bisect each other at point $E$ (i.e., $AE = EC$ and $DE=EB$).

Step2: Analyze $\triangle ADE$ and $\triangle CBE$

In $\triangle ADE$ and $\triangle CBE$, $\angle AED=\angle BEC$ (vertically - opposite angles), $AE = EC$, $DE = EB$. By the Side - Angle - Side (SAS) congruence criterion, $\triangle ADE\cong\triangle CBE$.

Step3: Analyze $\triangle ABC$ and $\triangle CDA$

In $\triangle ABC$ and $\triangle CDA$, $AB = DC$, $BC = AD$, and $AC=CA$ (common side). By the Side - Side - Side (SSS) congruence criterion, $\triangle ABC\cong\triangle CDA$.

Step4: Analyze $\triangle ABE$ and $\triangle CDE$

In $\triangle ABE$ and $\triangle CDE$, $\angle AEB=\angle DEC$ (vertically - opposite angles), $AE = EC$, $DE = EB$. By the SAS congruence criterion, $\triangle ABE\cong\triangle CDE$.

Step5: Analyze $\triangle ABD$ and $\triangle CDB$

In $\triangle ABD$ and $\triangle CDB$, $AB = DC$, $AD = BC$, $BD=DB$ (common side). By the SSS congruence criterion, $\triangle ABD\cong\triangle CDB$.

Answer:

A. $\triangle ADE$ and $\triangle CBE$
C. $\triangle ABC$ and $\triangle CDA$
D. $\triangle ABE$ and $\triangle CDE$
F. $\triangle ABD$ and $\triangle CDB$