Question

use the figure below to answer the following questions.
find the values of x and y.
\\( x = 50, y = 40 \\)
\\( x = 20, y = 7 \\)
\\( x = 10, y = 12 \\)
\\( x = 10, y = 8 \\)

Explanation:

Step1: Analyze angles on line RU

Angles \(5x^\circ\), \(4x^\circ\), and the right angle (since \(SU\perp SV\) or from the figure's right angle) sum to \(180^\circ\)? Wait, no, actually, \(5x + 4x+ 90 = 180\)? Wait, no, looking at the figure, \(5x\) and \(4x\) are adjacent to a right angle? Wait, no, the straight line \(RU\) has angles: \((10y + 10)^\circ\), then the angle between \(RT\) and \(SU\) is \(5x\), and then a right angle? Wait, no, let's re - examine. The angles around point \(S\) on the straight line \(RU\): the angle \((10y + 10)^\circ\), then the angle between \(RS\) and \(ST\) is \((10y + 10)^\circ\), then between \(ST\) and \(SU\) is \(5x^\circ\), and between \(SU\) and \(SV\) is \(4x^\circ\), and since \(SU\) is a straight line? Wait, no, \(RU\) is a straight line, so the sum of angles on one side of a straight line is \(180^\circ\). Also, notice that \(5x+4x = 9x\), and since there is a right angle (the square symbol), so \(5x + 4x+90=180\)? Wait, no, the right angle is between \(ST\) and \(SW\)? No, looking at the options, let's check the first equation for \(x\):

From the angles \(5x\) and \(4x\) and the right angle: Wait, actually, \(5x + 4x=90\) (because they are complementary to a right angle? Wait, no, if we consider that \(5x + 4x = 90\) (since the angle between \(ST\) and \(SV\) is \(90^\circ\)? Wait, let's solve for \(x\) first. If \(5x+4x = 90\) (because they form a right angle, as per the figure's right - angle symbol), then \(9x=90\), so \(x = 10\).

Step2: Solve for y

Now that \(x = 10\), we look at the straight line \(RU\). The angle \((10y + 10)^\circ\) and the angle \(5x^\circ\) (with \(x = 10\), so \(5x=50\)) and the right angle? Wait, no, the straight line \(RU\) has angles: \((10y + 10)^\circ+5x^\circ + 90^\circ=180^\circ\)? Wait, no, if \(x = 10\), then \(5x = 50\). So \((10y + 10)+50 + 90=180\)? Wait, no, \((10y + 10)+5x=90\)? Wait, no, let's use the straight line \(RU\). The sum of angles on a straight line is \(180^\circ\). So \((10y + 10)+5x + 4x=180\)? No, that can't be. Wait, when \(x = 10\), \(5x = 50\), \(4x = 40\). Now, the angle \((10y + 10)\) and \(5x\) are supplementary to the right angle? Wait, no, let's check the option where \(x = 10\). Then for \(y\):

We know that \((10y + 10)+5x=90\) (because they form a right angle with the other angle). Since \(x = 10\), \(5x = 50\). So \(10y+10 + 50=90\), \(10y+60 = 90\), \(10y=30\)? No, that's not right. Wait, maybe the angle \((10y + 10)\) and \(5x\) are equal to \(90\) when combined with the right angle? Wait, no, let's check the option \(x = 10\), \(y = 8\):

If \(x = 10\), then \(5x=50\), \(4x = 40\). Then \((10y + 10)\) should be equal to \(180-(50 + 90)=40\)? Wait, no, \(10y+10=80\) when \(y = 8\) (since \(10\times8 + 10=90\)? No, \(10\times8+10 = 90\). Then \(90+50=140\), which is not right. Wait, maybe the angle \((10y + 10)\) and \(5x\) are supplementary. So \((10y + 10)+5x=180\). If \(x = 10\), then \(5x = 50\), so \(10y+10 + 50=180\), \(10y=120\), \(y = 12\)? No, that's not matching. Wait, maybe the correct approach is:

From the figure, the angle \((10y + 10)\) and the angle \(5x\) are equal to \(90^\circ\) (since they are complementary to the right angle formed by \(4x\) and \(5x\)? Wait, no, let's check the option \(x = 10\), \(y = 8\):

If \(x = 10\), then \(5x=50\), \(4x = 40\). Then \((10y + 10)=10\times8 + 10=90\). Now, \(90+50 + 40=180\), which works because \(RU\) is a straight line. Ah, there we go. So the sum of \((10y + 10)\), \(5x\), and \(4x\) is \(180^\circ\) (since they are on a straight line \(RU\)).

So, substitute \(x = 10\) and \(y = 8\):

\(10y+10+5x + 4x=10\times8 + 10+5\times10+4\times10=80 + 10+50 + 40=180\), which satisfies the straight - line angle sum.

Answer:

\(x = 10,y = 8\) (the option is \(x = 10,y = 8\))