Question

in the following figure, ab || ed and bc = ce. explain why dc = ac.
choose the correct answer below.
a. △abc is congruent to △dec by saa. using cpctc, ac = dc. since the line - segments are congruent, ac = dc.
b. △abc is congruent to △dec by asa. using cpctc, ac = dc. since the line - segments are congruent, ac = dc.
c. △abc is congruent to △dec by sas. using cpctc, ac = dc. since the line - segments are congruent, ac = dc.
d. △abc is congruent to △dec by sss. using cpctc, ac = dc. since the line - segments are congruent, ac = dc.

Explanation:

Step1: Identify vertical angles

$\angle ACB$ and $\angle DCE$ are vertical - angles, so $\angle ACB=\angle DCE$.

Step2: Use given parallel lines

Since $AB\parallel ED$, $\angle BAC=\angle EDC$ (alternate - interior angles).

Step3: Use given side - equality

We are given that $BC = CE$.

Step4: Determine triangle congruence

By the Angle - Angle - Side (AAS) congruence criterion, $\triangle ABC\cong\triangle DEC$.

Step5: Apply CPCTC

Corresponding Parts of Congruent Triangles are Congruent (CPCTC). So, $AB = DE$.

Answer:

B. $\triangle ABC$ is congruent to $\triangle DEC$ by AAS. Using CPCTC, $AB = DE$. Since the line segments are congruent, $AB = DE$.