Chapter 12.4, Problem 20E

(a)

To determine

To Explain: Why $ABC$ is an isosceles triangle?

Answer to Problem 20E

  $AB=BC$

Explanation of Solution

Given:

  

Since $AB=BC$

Thus $ABC$ is an isosceles triangle.

(b)

To determine

To Explain: Why $\angle BAE\cong \angle BCE$ ?

Answer to Problem 20E

Base angle theorem, $AB=BC$

Explanation of Solution

Given:

  

Since $ABC$ is an isosceles triangle.

  $AB=BC$

Thus

  $\angle BAE\cong \angle BCE$

Using base angle theorem.

(c)

To determine

To Show: $\Delta ABE\cong \Delta CBE$ .

Explanation of Solution

Given:

  

In $\Delta ABEand\Delta CBE$

  $\begin{array}{l}AB=BC\\ \angle BAE\cong \angle BCE\\ BE=BE(commmon)\\ thus\\ \Delta ABE\cong \Delta CBE.\end{array}$

(d)

To determine

To Find: The measure of $\angle BAE.$ .

Answer to Problem 20E

  $\angle BAE={30}^{°}$

Explanation of Solution

Given:

  

  $ABCD$ is a rhombus formed by two equilateral triangles.

Thus

  $\angle BAD={60}^{°}$

  $AC$ is the bisector of it.

Thus

  $\begin{array}{l}\angle BAE=\frac{{60}^{°}}{2}\\ \angle BAE={30}^{°}\end{array}$