Answer
$x = 18$
$y = 108$
Work Step by Step
The kite pictured in this exercise has been divided into two triangles by its diagonal, with two sides marked congruent and one side that is shared by both triangles and, thus, represents another congruent side for the two triangles. Therefore, the two triangles are congruent by the SSS postulate.
Corresponding parts of congruent triangles are congruent.
We now turn our attention to the angles within each triangle. In one triangle, we already have expressions for two of the angles. We also have a value for the third angle because this third angle is $45^{\circ}$ because it is a corresponding angle, and, when added to the other corresponding angle, will equal $90^{\circ}$. We can now use the triangle sum theorem, which states that the sum of the interior angles of a triangle equal $180^{\circ}$.
$(\frac{3x}{2}) + (6x) + (45) = 180$
Subtract $45$ from each side to move constants to the right side of the equation:
$(\frac{3x}{2}) + (6x) = 135$
Let's get rid of the fraction by multiplying all terms by $2$:
$3x + 12x = 270$
Combine like terms:
$15x = 270$
Divide each side by $15$ to solve for $x$:
$x = 18$
Now, let's look at the congruent triangle. Plug in expressions for the corresponding angles from the other triangle into the triangle sum equation to find $y$:
$(\frac{3x}{2}) + (y) + (45) = 180$
Subtract $45$ from each side to move constants to the right side of the equation:
$(\frac{3x}{2}) + (y) = 135$
Let's get rid of the fraction by multiplying all terms by $2$:
$3x + 2y = 270$
Substitute $18$ for $x$:
$3(18) + 2y = 270$
Multiply first, according to order of operations:
$54 + 2y = 270$
Subtract $54$ from both sides of the equation to move constants to the right side of the equation:
$2y = 216$
Divide both sides by $2$ to solve for $y$:
$y = 108$
