Question

use the string art pattern below to find the values of x, y, and z

Explanation:

Step1: Analyze the right angle and given angle for \( x \)

We know that the right angle is \( 90^\circ \), and one of the angles adjacent to \( x \) is \( 86^\circ \). Since these two angles and \( x \) form a triangle (or a linear pair with the right angle? Wait, actually, looking at the diagram, the right angle and \( 86^\circ \) and \( x \) should satisfy \( x + 86^\circ= 90^\circ \)? Wait, no, maybe it's a triangle with a right angle. Wait, actually, the angle \( x \) and \( 86^\circ \) are complementary if they are in a right - angled triangle. So \( x=90 - 86 \)
\( x = 4^\circ \)

Step2: Analyze the triangle for \( y \)

We know that the sum of angles in a triangle is \( 180^\circ \). We have one angle as \( 101^\circ \) and we just found \( x = 4^\circ \). So \( y=180-(101 + 4) \)
\( y=180 - 105=75^\circ \)

Step3: Analyze the triangle for \( z \)

We know that the sum of angles in a triangle is \( 180^\circ \). We have one angle as \( 26^\circ \) and we know that the angle adjacent to \( y \) (since they are supplementary? Wait, no, looking at the diagram, the triangle with angle \( z \), \( 26^\circ \) and the angle supplementary to \( y \)? Wait, no, let's re - examine. Wait, we can also use the fact that the sum of angles in a triangle is \( 180^\circ \). If we consider the triangle where one angle is \( 26^\circ \), and we know that the other angle (let's say) is related to \( y \). Wait, actually, we can use the fact that the sum of angles in a triangle is \( 180^\circ \). Let's assume that in the triangle containing \( z \), \( 26^\circ \) and another angle. Wait, maybe a better way: we know that the sum of angles in a triangle is \( 180^\circ \). If we have a triangle with angles \( z \), \( 26^\circ \) and the angle that is supplementary to \( y \)? No, wait, let's use the fact that we can find \( z \) by considering the triangle. Wait, we know that \( y = 75^\circ \), and if we look at the triangle with angle \( z \), \( 26^\circ \) and the angle equal to \( 180 - y \)? No, maybe I made a mistake. Wait, let's start over.

Wait, the diagram seems to be a series of triangles formed by the strings. Let's consider the right - angled part first. For \( x \): since there is a right angle (\( 90^\circ \)) and an angle of \( 86^\circ \), then \( x=90 - 86 = 4^\circ \) (because they are complementary in the right - angled triangle).

For \( y \): In the triangle with angles \( 101^\circ \), \( x = 4^\circ \) and \( y \), we know that the sum of angles in a triangle is \( 180^\circ \). So \( y=180-(101 + 4)=75^\circ \)

For \( z \): In the triangle with angles \( 26^\circ \), and the angle supplementary to \( y \)? No, wait, in the triangle with angles \( z \), \( 26^\circ \) and the angle equal to \( 180 - y \)? Wait, no, let's consider the triangle where one angle is \( z \), one is \( 26^\circ \) and the third angle is equal to \( 180 - y \)? Wait, no, maybe the triangle has angles \( z \), \( 26^\circ \) and \( 180 - 101 - x \)? No, I think I messed up. Wait, another approach: the sum of angles in a triangle is \( 180^\circ \). Let's assume that in the triangle containing \( z \), we have angles \( z \), \( 26^\circ \) and \( 180 - 101 - 4=75^\circ \)? Wait, no, that can't be. Wait, maybe the triangle with \( z \) has angles \( z \), \( 26^\circ \) and \( 180 - 75=105^\circ \)? No, that's not right. Wait, I think I made a mistake in the first step. Wait, the right angle: maybe \( x \) and \( 86^\circ \) are supplementary? No, the diagram shows a right angle (a square corner), so \( x + 86^\circ=90^\circ \), so \( x = 4^\circ \) is correct.

Then, for the triangle with angle \( 101^\circ \), \( x = 4^\circ \) and \( y \), sum of angles is \( 180^\circ \), so \( y=180 - 101 - 4 = 75^\circ \)

Now, for \( z \): In the triangle with angle \( z \), \( 26^\circ \) and the angle that is equal to \( 180 - 75=105^\circ \)? No, that would make \( z=180 - 26 - 105 = 49^\circ \)? Wait, no, maybe the triangle has angles \( z \), \( 26^\circ \) and \( 180 - 101 - 4 = 75^\circ \)? No, that's the same as \( y \). Wait, maybe the correct way is:

We know that in a triangle, the sum of angles is \( 180^\circ \). Let's look at the triangle where one angle is \( z \), one is \( 26^\circ \) and the third angle is equal to \( 180 - 101 - x \). Wait, \( x = 4^\circ \), \( 180 - 101 - 4=75^\circ \). So \( z=180-(26 + 75)=180 - 101 = 79^\circ \)? Wait, no, \( 26+75 = 101 \), \( 180 - 101 = 79 \). So \( z = 79^\circ \)

Wait, let's verify:

For \( x \): Right angle is \( 90^\circ \), so \( x=90 - 86 = 4^\circ \)

For \( y \): Triangle with angles \( 101^\circ \), \( x = 4^\circ \), so \( y=180-(101 + 4)=75^\circ \)

For \( z \): Triangle with angles \( 26^\circ \), \( y = 75^\circ \)? No, that would be \( z=180-(26 + 75)=79^\circ \)

Yes, that makes sense.

Answer:

\( x = 4^\circ \), \( y = 75^\circ \), \( z = 79^\circ \)