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$72°$ .

Therefore,$∠ACD=∠ADC=72°$ .

Therefore,$∠FCD=∠ACD=72°$ .

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 $36°$.

Therefore, $∠ADF=∠CDF=36°$.

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

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

Therefore, it can be obtained that:

$\begin{array}{c}∠CFD+∠CDF+∠FCD=180°\\ ∠CFD+36°+72°=180°\\ ∠CFD+108°=180°\\ ∠CFD=180°−108°\\ ∠CFD=72°\end{array}$

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

In the parallelogram, both pairs of opposite angles are congruent.

Therefore,$∠ABC=∠ADC=72°$

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

In the parallelogram, both pairs of opposite sides are parallel.

As,$ABâˆ\yen CD$ and ACis a transversal.

Therefore, it can be seen that the angles $∠BAC$and $∠ACD$are the alternate interior angles.

Therefore,$∠BAC=∠ACD=72°$

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

$∠ADCâ‰\dots ∠ABCâ‰\dots ∠ACDâ‰\dots ∠CFDâ‰\dots ∠BAC$