Answer:
Segment PQ
Explanation:
To find the shortest segment, we must know the correct theorem to prove it.
According to an unnamed theorem, if two sides of a triangle are not congruent, then the larger angle lies across from the longest side.
Similarly in another theorem, if two angles of a triangle are not congruent then the longer side lies across from the largest angle.
Since we are given angles, we will use the second theorem to help guide us in knowing which segment of this entire diagram is the shortest.
As you can see, segment QS is a shared side with two triangles called ΔPQS and ΔHQS, or however you want to call these two triangles.
In triangle PQS, we can see that the smallest angle is 56 degrees. The side opposite to that angle is segment PQ, thus segment PQ is the shortest segment within triangle PQS.
Let's look as triangle HQS now. We can see that the smallest angle is 51 degrees, the side opposite to that smallest angle is segment QS(our shared side), keep this in mind. So, segment QS is the shortest side within triangle HQS.
After we have discovered the two shortest segments in each of these triangles, we have to determine the shortest side of this entire diagram according to the shared side/segment within these two triangles.
That being said, we know that the smallest side according to triangle HQS is QS (the shared side), all we have to find out now is the other angle in triangle PQS that has that same side and determine what kind of length it is (eg. next smallest, next next smallest(largest) side)- you'll see why.
In triangle PQS, the side that is being shared (segment QS) has a respective angle of 63 degrees. This would be the next smallest side in triangle PQS. However, in both these triangles, we ultimately want to find the most shortest side- in order to do that we need to look at triangle PQS and see which is the shortest side of that triangle, this side will be the side that is the shortest segment in all of this diagram- which was our main goal.
That shortest side is side PQ, a side that we have already found to be the shortest side of the triangle PQS. Thus, segment PQ is the shortest segment in triangle PQS and ΔQRS/aka the entire diagram.
In other words, if you ever come across a triangle with a shared side and it is up to you to determine the shortest segment, look for the shortest side in the triangle where the side is not the shared segment- this will be the shortest segment in all of the diagram.
I hope this detailed explanation helps!