Question

fill in the blank 4 points
what is the value of x and y in the following picture?

x = type your answer... y = type your answer

Explanation:

Step1: Use the property of trapezoid interior angles sum

The sum of interior angles of a quadrilateral is \(360^{\circ}\). In a trapezoid (assuming it is a right - trapezoid as one angle is \(90^{\circ}\)), we have the equation \(90+(2x)+138+(3y)=360\).
Simplify the left - hand side: \(2x + 3y+228 = 360\), then \(2x+3y=360 - 228=132\).
Also, since it is a trapezoid with two parallel sides, and we know one of the non - right angles adjacent to the right angle. If we assume the side with angle \(2x\) is adjacent to the right angle, and using the property of adjacent angles in a trapezoid (for non - parallel sides). But a more straightforward way: we know that in a trapezoid, if we consider the sum of angles. Another approach: since one angle is \(90^{\circ}\), and we can also use the fact that for the pair of angles \(2x\) (adjacent to \(90^{\circ}\)) and \(138^{\circ}\) and \(3y\). But using the sum of interior angles formula for a quadrilateral \(S=(n - 2)\times180^{\circ}\) (\(n = 4\), so \(S = 360^{\circ}\)).

Step2: Solve for \(x\)

We know that \(2x=90\) (because the angle adjacent to the right angle in a right - trapezoid, if we assume the non - parallel side and the right - angle side. Wait, no, using the sum formula \(90+(2x)+138+(3y)=360\). If we assume that the side with \(2x\) is a right - adjacent angle. Wait, no, correct way:

$$
LATEXBLOCK0
$$

Also, if we assume that \(2x = 90\) (if it is a right - trapezoid and \(2x\) is the angle adjacent to the right angle), then \(x = 45\). Substitute \(x = 45\) into \(2x+3y=132\), we get \(90+3y=132\).

Step3: Solve for \(y\)

$$
LATEXBLOCK1
$$

Answer:

\(x = 45\), \(y = 14\)