Question

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

Answer 1

Answer:

The measure of angle CAD is 83 degrees.

Step-by-step explanation:

Given information: ABCD is a parallelogram, AC is a diagonal, $\angle ABC=40^{\circ}$ and  $\angle ACD=57^{\circ}$.

The opposite sides of parallelogram are congruent.

The diagonal AC divides the parallelogram in two congruent triangles.

In triangle ABC and ADC,

$AB\cong CD$                                     (Opposite sides of parallelogram)

$\angle ABC\cong \angle ADC$         (Opposite angles of parallelogram)

$BC\cong DA$                                     (Opposite sides of parallelogram)

By SAS postulate,

$\triangle ABC\cong \triangle CDA$

Since we know that opposite angles of parallelogram are equal, therefore

$\angle ABC\cong \angle ADC$

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

According to the angle sum property the sum of interior angles of a triangle is 180 degrees.

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

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

$\angle CAD=83^{\circ}$

Therefore the measure of angle CAD is 83 degrees.