#### Answer

1. TWX is isosceles--given.
2. The angles of the bases of isosceles triangles are congruent, so X and W are congruent.
3. RY and WX are parallel.
4. Thus, angle W and angle WRY are supplementary, and angle X and angle XYR are supplementary.
5. Since it is a four-sided shape with opposite angles being supplementary and corresponding angles being congruent, it is a trapezoid.

#### Work Step by Step

To prove a quadrilateral is an isosceles trapezoid we have to prove that the two legs are congruent and base angles of the lower base segment are congruent and the upper base angles are congruent too.
1- in the triangle TWX, given that RY parallel to WX we conclude that $ \angle TRY= \angle RWX , \angle TYR = \angle YXW $ since the corresponding angles of two parallel lines are congruent.
2- in triangle TRW, and since the base angles are congruent so its an isosceles triangle such that TR = TY .
3- by segment addition TR+RW= TY+YX and since TR= TY therefore RW= YX.
4- given that angle W = angle X.
5- angle R and Y are a straight angles
$ \angle R= \angle TRY + \angle YRW = 180 $
$ \angle Y= \angle TYR + \angle RYX= 180 $
6- $ \angle TRY + \angle YRW= \angle TYR + \angle RYX $
Since we proved that $ \angle TRY= \angle TYR $
Therefore we conclude that $ \angle YRW = \angle RYX $ .
7- RYXW Is an isosceles trapezoid with pairs of congruent base angles and two congruent legs.