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

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)`.