Chapter 15, Problem 17RP

To determine

To Justify: The required statement in the given figure.

Explanation of Solution

Given:

  $\begin{array}{l}\overline{BC}\cong \overline{EC}\\ \angle A\cong \angle C\end{array}$

  

Formula Used:

SAS Congruence: It states that two triangles will congruent if two adjacent side and angle is equal.

Calculation:

  $\begin{array}{l}\overline{BC}\cong \overline{EC}\\ \angle A\cong \angle C\\ In\Delta ABC\\ \angle A\cong \angle C\\ \Rightarrow \overline{AB}\cong \overline{BC}\cong \overline{EC}\\ In\Delta BCE\\ \overline{BC}\cong \overline{EC}\\ \angle EBC\cong \angle BEC\\ \because \overline{AB}\cong \overline{BC}\cong \overline{EC}\\ and\angle C\text{is common}\\ \Delta ABC\cong \Delta BCE\\ \therefore \overline{BE}\cong \overline{AC}\\ In\Delta ABE\\ \angle BEA=\angle EBC+\angle ECB\\ \text{As shown }AB>AE\\ \Rightarrow EC>AE\end{array}$

Conclusion:

Hence, Proved.