Question

The parallelogram ABCD below, line AC is a diagonal, the measure of angle ABC is 40, and the measure of angle ACD is 57. What is the measure of angel of CAD?

Answer 1

Answer:

Using properties of parallelogram and angle sum property of a triangle in the figure as shown below in the attachment

In parallelogram ABCD, AC is a diagonal.

Given:  $\angle ABC = 40^{\circ}$ and $\angle ACD= 57^{\circ}$

As, we know that opposite angles in parallelogram are equal.

therefore,

$\angle ABC =\angle ADC= 40^{\circ}$

Now, in ΔADC

Sum of the measures of angles in a triangle is 180 degree.

$\angle ACD+ \angle ADC+\angle DAC =180^{\circ}$

Substituting the values of $\angle ADC= 40^{\circ}$ and  $\angle ACD= 57^{\circ}$  we have;

$57^{\circ}+40^{\circ}+\angle DAC =180^{\circ}$

or

$97^{\circ}+\angle DAC =180^{\circ}$

Subtract $97^{\circ}$ from both sides we get

$\angle DAC =180 -97 =83^{\circ}$

Therefore, the measure of angle CAD is $83^{\circ}$.