Question

In △PQR, PQ = PR and QR = 18 in. The altitude PN = 12 in and S is a point on the extension of QR so that PS = 20 in. Prove that ∠QPS is a right angle.

Answer 1

Focus on the triangle PQN and triangle SQP, prove that they are similar:
PQ=PR, PN is the altitude, so QN=NR=9
PQ of ΔSQP:QN=15:9=5:3
PS:PN=20:12=5:3
∠PQN=∠SQP (they are the same angle, reflexive property)
therefore, according to the SAS similarity theorem, ΔPQN is similar to ΔSQP
∠PNQ=∠SPQ =90 degree
therefore, QPS is a right angle.

you may have to reword it depending on what your teacher requires, column proof or paragraph proof.