Suppose we have angle PQR with P, Q, and R non-collinear, and ray QS distinct from ray QR such that angle PQS is congruent to angle PQR. Prove that if angle PQT is congruent to angle PQR, then either ray QT = ray QR or ray QT = ray QS.

Ariel Wilkinson 2022-10-01 Answered
Proof with congruence of angles
Suppose we have angle PQR with P, Q, and R non-collinear, and ray QS distinct from ray QR such that angle PQS is congruent to angle PQR. Prove that if angle PQT is congruent to angle PQR, then either ray Q T = ray QR or ray Q T = ray QS.
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Answers (1)

odejicahfc
Answered 2022-10-02 Author has 10 answers
Step 1
We'll assume given a line segment PQ, and three distinct rays QR, QS, and QT, each making the same angle x with PQ; we'll show this leads to a contradiction.
We'll assume x is not a right angle; we'll come back to deal with that case later.
There's a line L through P, parallel to QR. This line meets the ray QS at U, and it meets the ray QT at V (this is where we need the assumption that x is not a right angle). Q P U = P Q U = x, and Q P V = P Q V = x, so P U Q = P V Q. So line segments QU and QV make the same angle with line L, so these line segments are parallel. But that's impossible, since they meet at Q.
Step 2
Now if x is a right angle, then the line L through P parallel to QR can't meet either of the rays QS and QT --- if it did, you'd get a triangle with two right angles. So QR, QS, and QT are all rays through Q parallel to L. But that says there are (at least) two lines through Q parallel to L, which contradicts the parallel postulate, and we're done.
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