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

The measure of angle CAD is 83 degrees.

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.