Prove that AC = CE and AB = DE given that AB || DE and DC = CB.

Question

ScanSolve AI Tutor · Accepted Answer

To prove the statements, we will use the properties of parallel lines and congruent triangles.

Identify the triangles and given information.
We are given two triangles, $\triangle ABC$ and $\triangle EDC$ .
From the problem statement and diagram:
- $AB \parallel DE$ (indicated by the arrows on the lines).
- $DC = CB$ (indicated by the single tick marks on the lines).
Prove that $\triangle ABC$ is congruent to $\triangle EDC$ .
Consider $\triangle ABC$ and $\triangle EDC$ :
1. $\angle ACB = \angle ECD$ (These are vertically opposite angles).
2. $CB = DC$ (This is given in the problem statement).
3. $\angle ABC = \angle EDC$ (These are alternate interior angles because $AB \parallel DE$ and $BD$ is a transversal).
By the Angle-Side-Angle (ASA) congruence criterion, $\triangle ABC \cong \triangle EDC$ .
Use Corresponding Parts of Congruent Triangles are Congruent (CPCTC) to prove the required statements.
Since $\triangle ABC \cong \triangle EDC$ :
(1) The corresponding sides $AC$ and $CE$ must be equal.
Therefore, $\boxed{AC = CE}$ .

(2) The corresponding sides $AB$ and $DE$ must be equal.
Therefore, $\boxed{AB = DE}$ .

Answer