Chapter 4.6, Problem 11MRE

To determine

Prove $\Delta ABC$ is isosceles.

Explanation of Solution

Given:

  $\overline{BE}\cong \overline{CD}$

  $\overline{BD}\cong \overline{CE}$

Calculation:

The diagram is :

  

To prove that $\Delta ABC$ is isosceles , it is sufficient to prove that $\angle B\cong \angle C$ .

Consider the triangles $\Delta BEC\text{ and }\Delta CDB$ :

  $\overline{BE}\cong \overline{CD}$

  $\overline{BD}\cong \overline{CE}\Rightarrow \overline{EC}\cong \overline{DB}$

  $\overline{BC}\cong \overline{CB}$

So, by SSS Postulate , $\Delta BEC\cong \Delta CDB$

So, $\angle EBC\cong \angle DCB\Rightarrow \angle B\cong \angle C$

So, $\Delta ABC$ is isosceles

Hence proved.