Question

1. find the values of x and y.

Explanation:

Step1: Solve for \( x \)

The angles \( (5x - 17)^\circ \) and \( (3x - 11)^\circ \) are supplementary (they form a linear pair, so their sum is \( 180^\circ \)).
\[
(5x - 17) + (3x - 11) = 180
\]
Combine like terms:
\[
8x - 28 = 180
\]
Add 28 to both sides:
\[
8x = 180 + 28 = 208
\]
Divide both sides by 8:
\[
x = \frac{208}{8} = 26
\]

Step2: Solve for \( y \)

The angle \( (2y + 5)^\circ \) and the right angle ( \( 90^\circ \)) are complementary to the angle \( (5x - 17)^\circ \), or we can see that \( (2y + 5)^\circ \) and the right angle form a relationship with the linear pair. Wait, actually, since \( (5x - 17)^\circ \) and \( (3x - 11)^\circ \) are supplementary, and there's a right angle, so \( (2y + 5)^\circ \) should be such that when added to the right angle and the other angle? Wait, no, looking at the diagram, the angle \( (2y + 5)^\circ \) and the right angle ( \( 90^\circ \)) and the angle \( (5x - 17)^\circ \)? Wait, no, actually, the angle \( (2y + 5)^\circ \) and the right angle: Wait, no, the two angles \( (5x - 17)^\circ \) and \( (3x - 11)^\circ \) are on a straight line, so they sum to \( 180^\circ \). Then, the angle \( (2y + 5)^\circ \) is adjacent to the right angle, so \( (2y + 5)^\circ + 90^\circ + (3x - 11)^\circ = 180^\circ \)? Wait, no, maybe simpler: since \( (5x - 17)^\circ \) and \( (3x - 11)^\circ \) are supplementary, we found \( x = 26 \), so \( 5x - 17 = 5*26 - 17 = 130 - 17 = 113^\circ \), and \( 3x - 11 = 3*26 - 11 = 78 - 11 = 67^\circ \). Then, the angle \( (2y + 5)^\circ \) and the right angle ( \( 90^\circ \)) and \( 67^\circ \)? Wait, no, the right angle is between the horizontal line and the vertical line, so the angle \( (2y + 5)^\circ \) and \( 67^\circ \) should add up to \( 90^\circ \) (since they are complementary to the right angle? Wait, no, the right angle is \( 90^\circ \), so \( (2y + 5)^\circ + 67^\circ = 90^\circ \)? Wait, no, let's re-examine. The diagram has a right angle (the square), so the angle \( (2y + 5)^\circ \) and the angle \( (3x - 11)^\circ \) are complementary? Wait, no, the right angle is \( 90^\circ \), so \( (2y + 5)^\circ + (3x - 11)^\circ = 90^\circ \)? Wait, no, the straight line is \( 180^\circ \), so the right angle ( \( 90^\circ \)) plus \( (2y + 5)^\circ \) plus \( (3x - 11)^\circ \) should equal \( 180^\circ \)? Wait, no, that would be \( 90 + (2y + 5) + (3x - 11) = 180 \), which simplifies to \( (2y + 5) + (3x - 11) = 90 \), so \( 2y + 3x - 6 = 90 \), so \( 2y + 3x = 96 \). We know \( x = 26 \), so \( 3x = 78 \), so \( 2y + 78 = 96 \), so \( 2y = 96 - 78 = 18 \), so \( y = 9 \). Wait, let's check again.

Wait, the two angles \( (5x - 17)^\circ \) and \( (3x - 11)^\circ \) are supplementary, so:

\( 5x - 17 + 3x - 11 = 180 \)

\( 8x - 28 = 180 \)

\( 8x = 208 \)

\( x = 26 \). Correct.

Then, the angle \( (2y + 5)^\circ \) and the angle \( (3x - 11)^\circ \) are complementary to the right angle? Wait, the right angle is \( 90^\circ \), so the sum of \( (2y + 5)^\circ \) and \( (3x - 11)^\circ \) should be \( 90^\circ \), because they are adjacent to the right angle on a straight line? Wait, no, the straight line is \( 180^\circ \), so right angle ( \( 90^\circ \)) + \( (2y + 5)^\circ \) + \( (3x - 11)^\circ \) = \( 180^\circ \). So:

\( 90 + (2y + 5) + (3x - 11) = 180 \)

Simplify:

\( (2y + 5) + (3x - 11) = 90 \)

\( 2y + 3x - 6 = 90 \)

\( 2y + 3x = 96 \)

Substitute \( x = 26 \):

\( 2y + 3*26 = 96 \)

\( 2y + 78 = 96 \)

\( 2y = 18 \)

\( y = 9 \)

Yes, that makes sense.

Answer:

\( x = 26 \), \( y = 9 \)