Question

what are the values of x, y, and z? diagram with angles 66°, 45°, 30°, 43°, 119° and angles x°, y°, z° x=□, y=□, and z=□ (simplify your answers. do not include the degree symbol in your answer.)

Explanation:

Step1: Find x using triangle angle sum

In a triangle, the sum of angles is \(180^\circ\). For the triangle with angles \(x\), \(45^\circ\), and \(66^\circ + 30^\circ = 96^\circ\) (wait, no, correct triangle: angles \(x\), \(45^\circ\), and the third angle which is \(180 - (66 + 30 + 45)\)? Wait, no, let's look at the left triangle. The angles in the left triangle: \(x\), \(45^\circ\), and the angle adjacent to \(66^\circ\) and \(30^\circ\)? Wait, no, the left triangle has angles \(x\), \(45^\circ\), and the angle formed by \(66^\circ\) and \(30^\circ\)? Wait, no, actually, the sum of angles in a triangle is \(180\). So for the left triangle: \(x + 45 + (66 + 30)\)? No, wait, no. Wait, the two triangles share a vertical angle? Wait, the angle between \(30^\circ\) and \(43^\circ\) is vertical, so they are equal? Wait, no, vertical angles are equal. Wait, the angle opposite to \(30^\circ\) and \(43^\circ\)? Wait, maybe better to find \(x\) first.

In the left triangle, angles are \(x\), \(45^\circ\), and the angle that is \(180 - 66 - (30 + 45)\)? No, wait, let's calculate the angle at the bottom left. Wait, the triangle with angles \(x\), \(45^\circ\), and the angle adjacent to \(66^\circ\). Wait, the sum of angles in a triangle is \(180\). So \(x + 45 + (180 - 66 - 30 - 45)\)? No, I'm confused. Wait, let's do it properly.

First, in the triangle with angles \(66^\circ\), \(30^\circ\), and the third angle (let's call it \(A\)): \(A = 180 - 66 - 30 = 84^\circ\). Then, in the left triangle, angles are \(x\), \(45^\circ\), and \(A\)? No, wait, no. Wait, the left triangle has angles \(x\), \(45^\circ\), and the angle that is supplementary to \(A\)? No, maybe not. Wait, another approach: the sum of angles in a triangle is \(180\). So for the left triangle, angles are \(x\), \(45^\circ\), and the angle which is \(180 - 66 - (30 + 45)\)? No, let's look at the straight line. Wait, maybe the angle \(x\) is in a triangle where the other two angles are \(45^\circ\) and \( (180 - 66 - 30) \)? Wait, \(66 + 30 = 96\), so \(180 - 96 = 84\), then \(x + 45 + 84 = 180\)? No, \(45 + 84 = 129\), \(180 - 129 = 51\). So \(x = 51\).

Step2: Find z using linear pair

The angle \(z\) and \(119^\circ\) are supplementary (linear pair), so \(z = 180 - 119 = 61\).

Step3: Find y using triangle angle sum

In the right triangle, angles are \(y\), \(43^\circ\), and \(z = 61^\circ\). So \(y + 43 + 61 = 180\). \(43 + 61 = 104\), so \(y = 180 - 104 = 76\). Wait, no, wait, the right triangle: angles are \(y\), \(43^\circ\), and \(z\), and also the vertical angle? Wait, no, the vertical angle to \(30^\circ\) is \(43^\circ\)? Wait, no, vertical angles are equal, so the angle opposite to \(30^\circ\) is \(43^\circ\)? Wait, maybe I made a mistake. Wait, let's re-examine.

Wait, the angle adjacent to \(119^\circ\) is \(z\), so \(z = 180 - 119 = 61\) (correct, linear pair). Then, in the triangle with angles \(y\), \(43^\circ\), and \(z\), and the vertical angle to \(30^\circ\) is \(43^\circ\)? Wait, no, the angle between \(43^\circ\) and \(z\) is part of the triangle. Wait, the triangle with angles \(y\), \(43^\circ\), and \(z\): sum is \(180\). So \(y + 43 + 61 = 180\), so \(y = 180 - 104 = 76\). Wait, but let's check the left triangle again. The left triangle: angles \(x\), \(45^\circ\), and \( (66 + 30) \)? No, \(66 + 30 = 96\), then \(x + 45 + 96 = 180\)? \(45 + 96 = 141\), \(180 - 141 = 39\). Oh! I made a mistake earlier. So correct: the left triangle has angles \(x\), \(45^\circ\), and \(66^\circ + 30^\circ = 96^\circ\)? No, that can't be, because the sum would be more than 180. Wait, no, the triangle is formed by \(x\), \(45^\circ\), and the angle that is \(180 - 66 - 30 = 84^\circ\)? Wait, no, the angle at the bottom is \(45^\circ + 30^\circ = 75^\circ\)? No, I'm getting confused. Let's use the triangle angle sum properly.

Wait, the left triangle: angles are \(x\), \(45^\circ\), and the angle opposite to the angle that is \(66^\circ\) plus the vertical angle. Wait, the vertical angle to the angle between \(30^\circ\) and \(43^\circ\) is equal. So the angle between \(30^\circ\) and \(43^\circ\) is vertical, so they are equal? Wait, no, vertical angles are formed by two intersecting lines, so the angle opposite to \(30^\circ\) is \(43^\circ\)? That can't be, unless \(30 = 43\), which is not. So maybe the angle adjacent to \(30^\circ\) is \(43^\circ\) in the other triangle. Wait, let's start over.

1. Find \(x\):
In the left triangle, the three angles are \(x\), \(45^\circ\), and \( (180 - 66 - 30) \). Wait, \(66 + 30 = 96\), so \(180 - 96 = 84\). Then \(x + 45 + 84 = 180\)? \(45 + 84 = 129\), \(180 - 129 = 51\). Wait, but maybe the angle is \(66^\circ\), \(30^\circ\), and \(x\)'s triangle. Wait, no, the left triangle has angles \(x\), \(45^\circ\), and the angle that is \(180 - 66 - (30 + 45)\)? No, that's not right. Wait, the sum of angles in a triangle is 180. Let's look at the angles at the bottom: \(45^\circ\) and \(30^\circ\) are in the same vertex, so the angle in the left triangle at that vertex is \(45^\circ\), and the angle in the middle triangle is \(30^\circ\), and the angle in the right triangle is \(43^\circ\). Wait, maybe the middle triangle has angles \(66^\circ\), \(30^\circ\), and the third angle \(A\), so \(A = 180 - 66 - 30 = 84^\circ\). Then, the left triangle has angles \(x\), \(45^\circ\), and \(A = 84^\circ\)? No, \(x + 45 + 84 = 180\) → \(x = 51\). That seems correct.

2. Find \(z\):
The angle \(z\) and \(119^\circ\) are supplementary (they form a linear pair), so \(z = 180 - 119 = 61\).

3. Find \(y\):
In the right triangle, the angles are \(y\), \(43^\circ\), and \(z = 61^\circ\). Also, the vertical angle to \(30^\circ\) is \(43^\circ\)? No, wait, the angle in the right triangle at the vertex with \(43^\circ\) and \(z\) is part of the triangle. So the sum of angles in the right triangle is \(180\), so \(y + 43 + 61 = 180\) → \(y = 180 - 104 = 76\). Wait, but let's check with the other triangle. Wait, maybe the angle \(y\) is in a triangle with angles \(180 - x - 45\), but no. Wait, maybe I made a mistake in \(x\). Let's recalculate \(x\):

Left triangle: angles are \(x\), \(45^\circ\), and the angle that is \(180 - 66 - 30 = 84^\circ\)? No, that angle is adjacent to \(x\)'s triangle. Wait, the correct way: the sum of angles in a triangle is 180. So \(x + 45 + (66 + 30) = 180\)? No, \(66 + 30 = 96\), \(x + 45 + 96 = 180\) → \(x = 180 - 141 = 39\). Ah! Here's the mistake. The angle in the left triangle is \(45^\circ\), \(x\), and the angle that is \(66^\circ + 30^\circ\)? No, that's not possible. Wait, the two triangles (left and middle) share a side, and the angle at the bottom is \(45^\circ\) (left) and \(30^\circ\) (middle), so the total angle at that vertex is \(45 + 30 = 75^\circ\)? No, that's not how triangles work. Wait, maybe the left triangle has angles \(x\), \(45^\circ\), and the angle opposite to \(66^\circ\) in the middle triangle. Wait, the middle triangle has angles \(66^\circ\), \(30^\circ\), and \(84^\circ\) (as before). Then, the left triangle has angles \(x\), \(45^\circ\), and \(84^\circ\), so \(x = 180 - 45 - 84 = 51\). That's correct. Then \(z = 180 - 119 = 61\). Then, the right triangle has angles \(y\), \(43^\circ\), and \(z = 61^\circ\), so \(y = 180 - 43 - 61 = 76\). Wait, but let's check the linear pair for \(y\). No, \(y\) is in the triangle. Wait, maybe the angle \(y\) is adjacent to \(119^\circ\)? No, \(119^\circ\) is adjacent to \(z\). So \(z = 61\), correct. Then \(y = 180 - 43 - 61 = 76\), and \(x = 180 - 45 - (66 + 30)\)? No, \(66 + 30 = 96\), \(45 + 96 = 141\), \(180 - 141 = 39\). Wait, now I'm really confused. Let's use the exterior angle theorem or vertical angles.

Vertical angles: the angle between \(30^\circ\) and \(43^\circ\) are vertical, so they should be equal? But \(30 eq 43\), so that's not possible. Wait, maybe the diagram is a quadrilateral? No, it's two triangles with a common vertex. Wait, let's look at the given angles: \(45^\circ\), \(30^\circ\), \(66^\circ\), \(43^\circ\), \(119^\circ\).

Let's find \(x\) first:

In the left triangle, angles are \(x\), \(45^\circ\), and the angle that is \(180 - 66 - 30 = 84^\circ\) (since the middle triangle has angles \(66^\circ\), \(30^\circ\), and \(84^\circ\)). So \(x + 45 + 84 = 180\) → \(x = 180 - 129 = 51\).

Now \(z\): the angle \(z\) and \(119^\circ\) are supplementary, so \(z = 180 - 119 = 61\).

Now \(y\): in the right triangle, angles are \(y\), \(43^\circ\), and \(z = 61^\circ\). Also, the vertical angle to \(30^\circ\) is \(43^\circ\)? No, but the sum of angles in the right triangle is \(180\), so \(y + 43 + 61 = 180\) → \(y = 76\).

Wait, but let's check with the other triangle. The left triangle: \(x = 51\), \(45\), \(84\) (sum 180). Correct. Middle triangle: \(66\), \(30\), \(84\) (sum 180). Correct. Right triangle: \(y = 76\), \(43\), \(61\) (sum 180). Correct. And \(z = 61\), supplementary to \(119\) (61 + 119 = 180). Correct. So \(x = 51\), \(y = 76\), \(z = 61\).

Answer:

\(x = 51\), \(y = 76\), and \(z = 61\)