Answer:
The measure of angle CAD is 83 degrees.
Step-by-step explanation:
Given information: ABCD is a parallelogram, AC is a diagonal, [tex]\angle ABC=40^{\circ}[/tex] and [tex]\angle ACD=57^{\circ}[/tex].
The opposite sides of parallelogram are congruent.
The diagonal AC divides the parallelogram in two congruent triangles.
In triangle ABC and ADC,
[tex]AB\cong CD[/tex] (Opposite sides of parallelogram)
[tex]\angle ABC\cong \angle ADC[/tex] (Opposite angles of parallelogram)
[tex]BC\cong DA[/tex] (Opposite sides of parallelogram)
By SAS postulate,
[tex]\triangle ABC\cong \triangle CDA[/tex]
Since we know that opposite angles of parallelogram are equal, therefore
[tex]\angle ABC\cong \angle ADC[/tex]
[tex]\angle ADC=40^{\circ}[/tex]
According to the angle sum property the sum of interior angles of a triangle is 180 degrees.
[tex]\angle CAD+\angle ACD+\angle ADC=180^{\circ}[/tex]
[tex]\angle CAD+57^{\circ}+40^{\circ}=180^{\circ}[/tex]
[tex]\angle CAD=83^{\circ}[/tex]
Therefore the measure of angle CAD is 83 degrees.
