Proof with congruence of anglesSuppose 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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Proof with congruence of anglesSuppose 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.

odejicahfc · Accepted Answer

Step 1We&#039;ll assume given a line segment PQ, and three distinct rays QR, QS, and QT, each making the same angle x with PQ; we&#039;ll show this leads to a contradiction.We&#039;ll assume x is not a right angle; we&#039;ll come back to deal with that case later.There&#039;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&#039;s impossible, since they meet at Q.Step 2Now if x is a right angle, then the line L through P parallel to QR can&#039;t meet either of the rays QS and QT --- if it did, you&#039;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&#039;re done.

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.

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