Question

1. in parallelogram abcd below, (overline{ac}) is a diagonal, the measure of (angle abc) is (40^{circ}), and the measure of (angle acd) is 57°. what is the measure of (angle cad)?

a. (40^{circ})
b. (57^{circ})
c. (77^{circ})
d. (83^{circ})
e. (97^{circ})

Explanation:

Step1: Use the property of parallelogram

In parallelogram \(ABCD\), \(AB\parallel CD\). So, \(\angle BAC=\angle ACD = 57^{\circ}\) (alternate - interior angles).

Step2: Use the angle - sum property of a triangle

In \(\triangle ABC\), we know that \(\angle ABC = 40^{\circ}\), \(\angle BAC = 57^{\circ}\). But we want to find \(\angle CAD\).
Since \(AB\parallel CD\), in \(\triangle ACD\), we know that \(\angle ADC=\angle ABC = 40^{\circ}\) (opposite angles of a parallelogram are equal).
By the angle - sum property of a triangle (\(\angle CAD+\angle ACD+\angle ADC = 180^{\circ}\)), we substitute \(\angle ACD = 57^{\circ}\) and \(\angle ADC = 40^{\circ}\) into the formula.
\(\angle CAD=180^{\circ}-\angle ACD - \angle ADC\)
\(\angle CAD=180^{\circ}-57^{\circ}-40^{\circ}\)
\(\angle CAD = 83^{\circ}\)

Answer:

D. \(83^{\circ}\)