Question

in exercises 21 and 22, quad rstu is a parallelogram. find the values of x, y, a, and b. (hint: use alt. int. angles for number 21)
21.
parallelogram rstu with side rs = 6, ru = 9, angle at u is 80°, side st = b, side tu = a, angle at s is x°, angle at r is y°
22.
parallelogram rstu with side ru = 12, angle at u has 45° and 35°, side rs, diagonals intersecting, angle at r is x° with a segment, angle at s is y°, side st = b, side with length 9, etc.

Explanation:

Step1: Use the property of parallelogram (opposite angles are equal)

In a parallelogram \(RSTU\), \(\angle S=\angle U = 80^{\circ}\), \(\angle R=\angle T\). Also, adjacent angles are supplementary (\(\angle S+\angle R = 180^{\circ}\), \(\angle T+\angle U=180^{\circ}\)).
Since \(x\) and \(\angle U\) are opposite angles in a parallelogram, \(x = 80^{\circ}\).
Since \(y\) and \(\angle T\) are adjacent angles (\(\angle T = 100^{\circ}\) as \(\angle T=180 - 80\)), and \(y\) and \(\angle T\) are adjacent angles in a parallelogram, \(y=100^{\circ}\)

Answer:

\(x = 80^{\circ}\), \(y = 100^{\circ}\)