$\triangle{ABC}$ such that $AB = AC$. $P$ lies on $\overline{AC}$ and $Q$ lies on $\overline{BC}$ such that $AP = AQ$. Find $m\angle{PQC}$ if $m\angle{BAQ}=30^\circ$.
I constructed the triangle:
Using the fact that all sides of the triangle equal to $180^\circ$, I found the equation
$$x+x+30+\angle{QAC}=180$$
$$\angle{QAC}= 180-2y$$
So
$$15=y-x$$
I tried to find a second equation using the sum of exterior angles, but I kept on resolving equation to the aforementioned one.
I feel that this construction is unique, but I'm unable to find a second equation to solve for either x or y.
EDIT:
I realized that $\angle{PQC}$ is $y-x$. Since we know that $y-x=15$ from the previous equation, we know that this is the answer.
My new question:
Is it possible to find the values of $x$ and $y$ explictily?
Best Answer
Note: I have deleted the proof. Since OP doesn't want it.
Claim
There are infinitely many triangles satisfying the given properties.
Proof
Take an arbitrary acute angle $x$. Construct the isosceles triangle $ABC$ with angles $x,x,90-2x$. Choose the point $Q$ on $BC$ such that $\angle BAQ = 30^\circ$. Now choose the point $P$ on $AC$ such that $|AQ| = |AP|.$ According to proof of OP we have $\angle PQC = 15^\circ$. Clearly we were able to construct to triangle for an arbitrary $x$. Therefore there can be infinitely many triangles.
Plot of two triangle with large and small $x$
Note: I have misprinted the point $P$ and $Q$ in the first image. Sorry.