91Ó°ÊÓ

The given diagram is:

It is being given that $\stackrel{¯}{AD}â‰\dots \stackrel{¯}{AC}$.

The angles opposite to the equal sides are also equal.

As,$\stackrel{¯}{AD}â‰\dots \stackrel{¯}{AC}$ , therefore, the angles opposite to the sidesAD and ACare also equal.

That implies,$∠ACD=∠ADC$ .

As the sum of the angles of the triangle is$180°$ .

Therefore, the sum of the angles of the triangle$ACD$ is$180°$ .

Therefore, it can be obtained that:

$\begin{array}{c}∠DAC+∠ACD+∠ADC=180°\\ 36°+∠ADC+∠ADC=180°\\ 2∠ADC=180°−36°\\ 2∠ADC=144°\\ ∠ADC=\frac{144°}{2}\\ ∠ADC=72°\end{array}$

Therefore, the measure of the angle$∠ADC$ is$72°$ .

It is also being given that$∠ADFâ‰\dots ∠CDF$.

Therefore,$∠ADF=∠CDF$ .

From the diagram it can be noticed that:

$∠ADC=∠ADF+∠CDF$.

Therefore, it can be obtained that:

$\begin{array}{c}∠ADC=∠ADF+∠CDF\\ 72°=∠ADF+∠ADF\\ 72°=2∠ADF\\ \frac{72°}{2}=∠ADF\\ 36°=∠ADF\end{array}$

Therefore, the measure of the angle$∠ADF$ is $36°$.

It is given that$m∠DAC=36$ .

From the given diagram it can be seen that$m∠DAC=m∠DAF=36$ .

Therefore, it can be seen that$∠ADF=∠DAF=36°$

In the triangle $△ADF$, it can be seen that$∠ADF=∠DAF=36°$.

Therefore, the two angles of the triangle are equal and so the triangle is an isosceles triangle.

$â–³ADF$is an isosceles triangle.