Circle geometry problem (find an angle in a figure)

Question

I’m trying to do this problem, which is to find the shaded angle.

I noticed that one of the triangles is isosceles, and so I could calculate the other two angles. And so using a result about alternate angles I think, I was able to calculate one of the angles of the triangle of interest.

The solution claims to use the angles corresponding to the same segment are equal, but I’m not sure how to see this.

Answer 1

As you note, $m\angle R=\frac{180-72}{2}=54^\circ$. Since $\angle R$ and $\angle U$ are inscribed angles that subtend the same arc, they must be congruent. Therefore $m\angle U=54^\circ$,

Answer 2

Look at angle VTU. It intercepts the same arc length as the 72 degree angle, which means that VTU has measure 72 degrees. In addition, note that Angle RVS and Angle UVT are equal because they are vertical angles. And remember that Angle RVS + Angle SRV = 108 degrees, and Angle RVS = Angle SRV because triangle RVS is isoceles (and angles opposite equal sides in an isoceles triangle are equal).

Therefore Angle RVS = 54 degrees, and that means Angle UVT = 54 degrees. Remember that angle VUT = 180 - UVT - VTU, or what we now know to be: 180 - 54 - 72 = 54 degrees.