Question

abdc is a trapezoid with \\( \overline{ab} \parallel \overline{cd} \\). find the measures of angles x, y, and z. angle x measures \\( \square^\circ \\) angle y measures \\( \square^\circ \\) angle z measures \\( \square^\circ \\)

Explanation:

Step1: Find angle x

Since \( AB \perp BD \) (right angle at B) and \( AB \parallel CD \), \( CD \perp BD \), so \( \angle x + 51^\circ = 90^\circ \).
\( \angle x = 90^\circ - 51^\circ = 39^\circ \)

Step2: Find angle y

In triangle \( ABD \), it's a right triangle ( \( \angle B = 90^\circ \) ), so \( \angle y + \angle x + 90^\circ = 180^\circ \).
We know \( \angle x = 39^\circ \), so \( \angle y = 180^\circ - 90^\circ - 39^\circ = 51^\circ \)

Step3: Find angle z

In trapezoid \( ABDC \), \( AB \parallel CD \), so \( \angle C + \angle z + \angle y = 180^\circ \) (consecutive interior angles).
\( \angle C = 45^\circ \), \( \angle y = 51^\circ \), so \( \angle z = 180^\circ - 45^\circ - 51^\circ = 84^\circ \)

Answer:

Angle \( x \) measures \( 39^\circ \)
Angle \( y \) measures \( 51^\circ \)
Angle \( z \) measures \( 84^\circ \)