Step by step explanation of finding the value of x and y

Step-by-step explanation:

In order to solve this problem, the easiest approach is to draw a line connecting the vertex between the side with a measure of (y) and (3) to the side with the length (x). This side should be parallel to the side with a length of (4).

Since the new side constructed (call this side (b)), is parallel to the side with a length of (4) and is intersected by segments that are perpendicular to the side with a length of (4), the side (b) and (4) must have the same measure. Therefore (b = 4).

The triangle formed with sides (x) and (b) is a (45 - 45 - 90) triangle. Meaning that its angles have the following measure, (45 - 45- 90). This means that its sides follow the following ratio:

angle : side opposite

Apply this logic to the given situation.

The side (b) divides the side (y) into two parts. Call the part that is a side of the triangle () the side that is a part of the rectangle is called (). As per the logic used to find the lengths of side (b), one can use similar logic to find the length of side (). The sides () and (3) are parallel and are intersected by the sides with a measure of (4) and (b), therefore  () and (3) are cognrunet. Thus ( = 3).

The base angles converse theorem states that when two angles in a triangle are congruent, the triangle is isosceles, and the sides opposite the congruent angles are also congruent. Thus, one can state the following:

Now add up the measurement of () and () to find the measure of (y):