To find the values of x and y in the given rhombus ABCD, we need to understand the properties of a rhombus. A rhombus is a quadrilateral with all sides of equal length.

- The diagonals of a rhombus bisect each other at right angles. This means that the diagonals intersect at a 90-degree angle.
- The diagonals of a rhombus divide it into four congruent triangles.
- The opposite angles in a rhombus are equal.

Given these properties, let's solve for x and y.

First, let's consider the diagonals of the rhombus. Let the intersection point of the diagonals be O.

- The diagonals AC and BD bisect each other at right angles, so angle AOC and angle BOD are right angles.

Since the diagonals divide the rhombus into congruent triangles, we can focus on one of these triangles, such as triangle AOB.

- In triangle AOB, angle AOB is equal to 90 degrees (as it is a right angle).
- The opposite angles in a rhombus are equal, so angle AOB is also equal to angle ABO.

Since the sum of angles in a triangle is 180 degrees, we can write the following equation:

angle AOB + angle ABO + angle BAO = 180 degrees

We know that angle AOB = 90 degrees, and angle AOB = angle ABO, so we can rewrite the equation as:

90 degrees + angle AOB + angle BAO = 180 degrees

Simplifying further, we get:

2 * angle AOB + angle BAO = 180 degrees

Since angle AOB = angle ABO, we can substitute angle AOB with x and angle BAO with y:

2x + y = 180 degrees

Now, let's consider the fact that opposite angles in a rhombus are equal. Since angle AOC is a right angle, angle AOD is also a right angle.

- The sum of angles in a triangle is 180 degrees, so we can write:

angle AOD + angle ADO + angle OAD = 180 degrees

Since angle AOD = 90 degrees and angle ADO = angle OAD, we can rewrite the equation as:

90 degrees + angle ADO + angle OAD = 180 degrees

Simplifying further, we get:

2 * angle ADO + angle OAD = 180 degrees

Substituting angle ADO with x and angle OAD with y, we get:

2x + y = 180 degrees

This equation is the same as the one we obtained earlier. Therefore, we can conclude that x = y.

Now, let's consider the fact that the diagonals of a rhombus bisect each other at right angles. This means that the diagonals AC and BD are perpendicular.

Since opposite sides of a rhombus are parallel, the diagonals bisect each other into two congruent right-angled triangles. In each of these triangles, the sum of angles is 180 degrees. Therefore, we can write:

angle ADC + angle ACB + angle CDB = 180 degrees

Since angle ADC = angle ACB (as opposite angles are equal), we can rewrite the equation as:

2 * angle ADC + angle CDB = 180 degrees

Substituting angle ADC with x and