Question

try it! 3. b. what are the values of x and y in the figure? 42° y° x° 57° 59° enter your answer.

Explanation:

Step1: Find x using triangle angle sum

In a triangle, the sum of angles is \(180^\circ\). For the triangle with \(57^\circ\) and \(59^\circ\):
\(x + 57 + 59 = 180\)
\(x + 116 = 180\)
\(x = 180 - 116 = 64\)? Wait, no, wait—wait, vertical angles? Wait, no, first, the two triangles have vertical angles, so the angle opposite \(x\) in the first triangle is equal to \(x\). Wait, no, let's correct.

Wait, the two triangles are vertical (share a common vertex, so vertical angles are equal). So first, find \(x\) in the right triangle (with \(57^\circ\) and \(59^\circ\)):

Sum of angles in a triangle: \(57 + 59 + x = 180\)? Wait, no, wait, the angles in the triangle are \(57^\circ\), \(59^\circ\), and the third angle (which is vertical to the angle in the left triangle). Wait, no, actually, the left triangle has angles \(42^\circ\), \(y^\circ\), and the vertical angle to \(x^\circ\). The right triangle has angles \(57^\circ\), \(59^\circ\), and \(x^\circ\) (since vertical angles are equal? Wait, no, vertical angles are equal, so the angle opposite \(x\) in the left triangle is equal to \(x\). Wait, let's re-express:

In the right triangle: angles are \(57^\circ\), \(59^\circ\), and the angle at the vertex (let's call it \(z\)), so \(57 + 59 + z = 180\). Then \(z = 180 - 57 - 59 = 64\). But \(z\) and \(x\) are vertical angles? Wait, no, \(x\) is the angle at the intersection, so actually, \(x\) is equal to the angle opposite in the other triangle. Wait, no, the two triangles are formed by intersecting lines, so the vertical angles are equal. So the angle in the left triangle (at the intersection) is equal to \(x\), and the angle in the right triangle (at the intersection) is also \(x\). Wait, no, let's do it properly.

First, find \(x\) in the right triangle: the three angles are \(57^\circ\), \(59^\circ\), and \(x\)? No, wait, no—wait, the figure is two triangles with a common vertex (intersecting lines), so the vertical angles are equal. So the angle in the left triangle (between \(42^\circ\) and \(y^\circ\)) is equal to the angle in the right triangle (between \(57^\circ\) and \(59^\circ\))? No, wait, no. Wait, the sum of angles in a triangle is \(180^\circ\). So for the right triangle: angles are \(57^\circ\), \(59^\circ\), and the third angle (let's call it \(A\)): \(57 + 59 + A = 180\) → \(A = 180 - 57 - 59 = 64\). Then, the angle \(A\) and \(x\) are vertical angles? No, \(x\) is the angle at the intersection, so actually, \(x\) is equal to \(A\)? Wait, no, maybe I mixed up. Wait, the two triangles: left triangle has angles \(42^\circ\), \(y^\circ\), and \(x^\circ\) (since the intersection angle is \(x\)), and right triangle has angles \(57^\circ\), \(59^\circ\), and \(x^\circ\) (because vertical angles are equal). Wait, that makes sense! Because the two triangles share the vertical angle \(x\), so the left triangle: \(42 + y + x = 180\), right triangle: \(57 + 59 + x = 180\). Wait, no, that can't be, because then \(x\) would be the same in both, but let's check the right triangle first.

Right triangle: \(57 + 59 + x = 180\) → \(116 + x = 180\) → \(x = 64\)? Wait, no, that's not right. Wait, no, the right triangle's angles are \(57^\circ\), \(59^\circ\), and the angle opposite to the left triangle's angle. Wait, I think I made a mistake. Let's start over.

The two triangles are formed by two intersecting lines, so the vertical angles are equal. Let's denote the vertical angles as \(x\) (the angle at the intersection). So in the right triangle, the three angles are \(57^\circ\), \(59^\circ\), and the angle adjacent to \(x\)? No, no. Wait, the sum of angles in a triangle is \(180^\circ\). So for the right triangle: angles are \(57^\circ\), \(59^\circ\), and the third angle (let's call it \(B\)): \(57 + 59 + B = 180\) → \(B = 180 - 57 - 59 = 64\). Then, \(B\) and \(x\) are vertical angles? No, \(x\) is the angle at the intersection, so \(x\) is equal to \(B\)? Wait, no, \(x\) is the angle between the two sides, so actually, \(x\) is equal to the angle in the left triangle's intersection. Wait, maybe the left triangle has angles \(42^\circ\), \(y^\circ\), and \(x^\circ\), and the right triangle has angles \(57^\circ\), \(59^\circ\), and \(x^\circ\) (because vertical angles are equal). Wait, that would mean that \(42 + y + x = 180\) and \(57 + 59 + x = 180\). Let's solve the right triangle first: \(57 + 59 + x = 180\) → \(116 + x = 180\) → \(x = 64\). Then, in the left triangle: \(42 + y + 64 = 180\) → \(106 + y = 180\) → \(y = 74\). Wait, but that doesn't seem right. Wait, no, maybe the vertical angles are the ones opposite, so the angle in the left triangle (between \(42^\circ\) and \(y^\circ\)) is equal to the angle in the right triangle (between \(57^\circ\) and \(59^\circ\))? No, that angle in the right triangle is \(180 - 57 - 59 = 64\), so the left triangle's angle (at the intersection) is also \(64^\circ\), so \(x = 64\)? Wait, no, \(x\) is the angle at the intersection, so \(x\) is equal to that angle. Then, in the left triangle, angles are \(42^\circ\), \(y^\circ\), and \(x^\circ\) (which is \(64^\circ\))? Wait, no, that would make the sum \(42 + y + 64 = 180\) → \(y = 74\). But let's check again.

Wait, maybe the correct approach is: the two triangles are vertical, so the angle opposite \(x\) in the left triangle is equal to \(x\). Wait, no, vertical angles are equal, so the angle at the intersection (x) is equal to the angle in the other triangle's intersection. Then, for the right triangle: angles are \(57^\circ\), \(59^\circ\), and \(x\)? No, that can't be, because \(57 + 59 + x = 180\) would mean \(x = 64\), and then the left triangle: \(42 + y + x = 180\) → \(y = 74\). But let's verify with the triangle angle sum.

Wait, another way: the sum of angles in a triangle is \(180^\circ\). So in the right triangle, angles are \(57^\circ\), \(59^\circ\), and the third angle (let's call it \(C\)): \(C = 180 - 57 - 59 = 64^\circ\). Then, \(C\) and \(x\) are vertical angles, so \(x = 64^\circ\). Then, in the left triangle, angles are \(42^\circ\), \(y^\circ\), and \(x^\circ\) (since \(x\) is the vertical angle), so \(42 + y + 64 = 180\) → \(y = 180 - 42 - 64 = 74\). Wait, but that seems correct. Wait, but maybe I mixed up the angles. Wait, no, the figure is two triangles with a common vertex (intersecting lines), so the vertical angles are equal. So the angle at the intersection (x) is equal to the angle in the other triangle's intersection. So the right triangle has angles 57, 59, and x? No, no, the right triangle's angles are 57, 59, and the angle opposite to the left triangle's angle. Wait, I think the correct answer is x = 64? No, wait, no—wait, 57 + 59 is 116, 180 - 116 is 64, so x is 64? Then y: 180 - 42 - 64 = 74? Wait, but let's check again.

Wait, maybe the left triangle's angles are 42, y, and the vertical angle of (57 + 59)? No, no. Wait, the sum of angles in a triangle is 180. So for the right triangle: 57 + 59 + x = 180 → x = 64. For the left triangle: 42 + y + x = 180 → y = 180 - 42 - 64 = 74. Yes, that makes sense.

Step2: Find y using triangle angle sum

Now that we have \(x = 64^\circ\), use the left triangle:
\(42 + y + x = 180\)
Substitute \(x = 64\):
\(42 + y + 64 = 180\)
\(y + 106 = 180\)
\(y = 180 - 106 = 74\)

Answer:

\(x = 64\), \(y = 74\)