Question

act prep practice! you must show complete work on your worksheet! answers that are only circled will not receive credit!
trapezoid abcd is isosceles with \\( \angle bdc \\) measuring \\( 23 ^ { \circ } \\) and \\( \angle acb \\) measuring \\( 50 ^ { \circ } \\). what is the measure of \\( \angle cbd \\)?
107
123
96
84
57

Explanation:

Step1: Properties of isosceles trapezoid

In an isosceles trapezoid \(ABCD\), \(AD = BC\) and \(\angle ADC=\angle BCD\). Also, \(AC = BD\) (diagonals of isosceles trapezoid are equal).

Step2: Use triangle angle - sum property

In \(\triangle BCD\), we know that the sum of angles in a triangle is \(180^{\circ}\). Let \(\angle CBD=x\). We are given \(\angle BDC = 23^{\circ}\) and \(\angle BCD=\angle ACB+\angle ACD\). Since \(AC = BD\) and \(AD = BC\), \(\angle ACD=\angle BDC = 23^{\circ}\) (by congruent triangles or property of isosceles trapezoid diagonals). So \(\angle BCD=50^{\circ}+23^{\circ}=73^{\circ}\).
Using the angle - sum property of a triangle \(\angle CBD+\angle BDC+\angle BCD = 180^{\circ}\).
Substitute the values: \(x + 23^{\circ}+73^{\circ}=180^{\circ}\).

Step3: Solve for \(x\)

$$
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Answer:

\(84^{\circ}\)