A trapezoid midsegment connects the midpoints of the two congruent sides of the trapezoid, and is parallel to the pair of parallel sides.
The length of the midsegment is the sum of the two bases divided by 2. Remember that the bases of a trapezoid are the two parallel sides.
To find angles within a trapezoid, remember that since there are two sides are parallel, the other sides can be seen as transversals, forming corresponding angles and same side interior angles. Using what is known about corresponding and same sider interior angles, it is possible to find the measures of missing angles in the trapezoid.
In this problem we’re being asked to find the length of this segment and the measure of these two angles. Well let’s start off by saying what do we know about this problem?
Well I see that this point is the midpoint of this side and this point is the midpoint of this other side. Since I have two parallel sides, it’s going to be a trapezoid and since these are the midpoints, what I’ve done is I’ve created a midsegment and the two key things that I know about midsegments, the first is that it’s parallel to the two bases, so I’m going to go back here and I’m going to mark this midsegment as being parallel, and I also know that the length of my midsegment is the average of the two bases, so if you added up the two bases and divided by 2, you would have the length of your midsegment.
So let’s find a first. 'a' is the distance of our midsegment, so I’m going to say a is equal to the average of your two bases which are 10 and 18. So 10 and 18 is 28, so a equals 28 divided by 2, so a is 14 and our units here are centimetres, so I’m going to write a is 14cm.
Now let’s find x. Well since these two lines are parallel, I can think of this side right here as a transversal creating corresponding angles which are always congruent. So x is congruent to 55 degrees because corresponding angles must be congruent. Now to find y, I’m going to have to look at this side as a transversal.
A couple of ways you could figure this out. The first way is to say well 120 degrees corresponds to this angle right here, so this angle must be 120 degrees. 120 degrees and y are on the same side of the transversal and it’s in between the two parallel lines, so those are same side which means y plus 120 degrees must be supplementary, so if I subtract 120 degrees, then I see that y must be 60 degrees.
So the two key things about solving this problem, one was remembering that a midsegment and a trapezoid is parallel to the two bases and its length is equal to the sum of the two bases divide by 2.