The Correct Answer and Explanation is:
The correct answer is 72 degrees.
The problem asks for the measure of angle TUB. Based on the provided diagram and labels, it is highly likely there is a typographical error in the question and it should be asking for the measure of angle TUS. The solution proceeds with this assumption.
To find the measure of angle TUS, we must first determine the measures of the other two angles within triangle STU, which are angle STU and angle TSU. The sum of angles in any triangle is 180 degrees.
First, we are given that the measure of angle VTU is 52 degrees. Since points S, T, and V are all on the same straight line, angle STU is the same as angle VTU. Therefore, the measure of angle STU is 52 degrees.
Next, we need to find the measure of angle TSU. This angle is the same as angle RSW. We can find this by first analyzing the smaller triangles formed by the intersection at point W. Consider triangle QRW. We are given that the measure of angle RQX, which is the same as angle RQW, is 46 degrees. The angle RWS is vertically opposite to angle SWU, so it has the same measure, which is 85 degrees. According to the Exterior Angle Theorem, the exterior angle of a triangle is equal to the sum of its two opposite interior angles. For triangle QRW, angle RWS is an exterior angle. Thus, angle RWS equals the sum of angle RQW and angle QRW.
Using this theorem:
85° = 46° + measure of angle QRW
Measure of angle QRW = 85° – 46° = 39°
Now consider triangle RWS. The sum of its angles is 180 degrees. We know angle SWR is 85 degrees and angle WRS (which is the same as angle QRW) is 39 degrees. We can find angle RSW:
Measure of angle RSW = 180° – 85° – 39° = 56°
Since angle TSU is the same as angle RSW, the measure of angle TSU is 56 degrees.
Finally, we return to triangle STU. We have angle STU = 52° and angle TSU = 56°. We can now calculate the target angle, TUS:
Measure of angle TUS = 180° – (52° + 56°) = 180° – 108° = 72°.
The given information that the measure of angle SXQ is 16 degrees is not required to solve the problem.
