Chapter 5, Problem 22CUR

To determine

To complete: $△ADF$ is an ___________ triangle.

Answer to Problem 22CUR

Isosceles

Explanation of Solution

Given:

The diagram:

  $\overline{AD}\cong \overline{AC}$

  $\overline{AE}\cong \overline{EC}$

  $\angle ADF\cong \angle CDF$

  $m\angle DAC=36$

Concept Used:

Theorem 4-1: Opposite angles of equal sides are equal in triangle.

Theorem 3-11: The addition of all angles of a triangle is 180.

Consider the figure shown below:

Consider the triangle $△ADC$ ,

Here $\overline{AD}\cong \overline{AC}$ , so by theorem 4-1, the angles $\angle ADC$ and $\angle ACD$ are congruent.

That is,

  $m\angle ADC=m\angle ACD$

Now by theorem 3-11,

  $\begin{array}{c}m\angle ADC+m\angle ACD+m\angle DAC=180\\ m\angle ADC+m\angle ADC+36=180\\ 2m\angle ADC+36=180\\ 2m\angle ADC=180-36\end{array}$

  $\begin{array}{c}2m\angle ADC=144\\ m\angle ADC=\frac{144}{2}\\ m\angle ADC=72\end{array}$

Now, it is given that $\angle ADF\cong \angle CDF$ ,

So,

  $m\angle ADF=m\angle CDF$

Consider,

  $\begin{array}{c}m\angle ADC=72\\ m\angle ADF+m\angle CDF=72\\ m\angle ADF+m\angle ADF=72\end{array}$

  $\begin{array}{c}2m\angle ADF=72\\ m\angle ADF=\frac{72}{2}\\ m\angle ADF=36\end{array}$

Now, in the triangle $△ADF$ , it can be observed that $m\angle ADF=36$ and $m\angle DAF=36$ .

So, the two angles $\angle ADF$ and $\angle DAF$ are congruent.

A triangle whose two angles are congruent is called an isosceles triangle.

Hence, the triangle $△ADF$ is an isosceles triangle.