Chapter 4.4, Problem 34WE

To determine

To find: A fact about $\Delta DEF$ shown in the figure such that $\Delta ABC$ is equilateral and $\angle CAD=\angle ABE=\angle BCF$ .

Answer to Problem 34WE

  $\Delta DEF$ is an equilateral triangle.

Explanation of Solution

Given:

In the figure shown, $\Delta ABC$ is equilateral and $\angle CAD=\angle ABE=\angle BCF$ .

  

Calculation:

In $\Delta ABC$ ,

  $\begin{array}{l}AB=AC=BC[\text{Given}]\\ \angle ABE=\angle CAD=\angle BCF[\text{Given}]\\ \angle BAE=60°-\angle CAD\\ \angle ACD=60°-\angle BCF\\ \angle CBF=60°-\angle ABE\end{array}$

So, $\angle BAE=\angle ACD=\angle CBF$

Therefore, by ASA Postulate, $\Delta ABE\cong \Delta ADC$ .

Similarly, $\Delta BCF\cong \Delta ABE\cong \Delta ADC$ .

Using property of congruent triangles.

  $\angle AEB=\angle BFC=\angle CDA$

Here, $\angle AEB,\angle BFC$ and $\angle CDA$ are the exterior angles of the triangle DEF .

Angles $\angle FED=\angle EDF=\angle DFE$ are also equal because these angles are supplement to the exterior angles $\angle AEB,\angle BFC$ and $\angle CDA$ .

Since, angles are congruent and their measure is $60°$ , therefore $\Delta DEF$ is equilateral triangle.