Prove the Following:

Question

A, B, and C lie in plane E. P and Q are on opposite sides of E, with P – A – Q. Given that line AC and line AB are perpendicular bisectors of segment PQ, prove that angle BPC is congruent to angle BQC.

Answer 1

From the given perpendicular bisectors, PA = QA

and PAB, QAB, PAC, QAC are all right angles

BA=AB and AC=CA by reflexive..

right triangles PAB and QAB are congruent by LL/SAS.

right triangles PAC and QAC are congruent by LL/SAS.

angles BPA and BQA are congruent by CPCTC

angles CPA and CQA are congruent by CPCTC

angle BPC = angle BPC + angle CPA <--- angle addition postulate

= angle BQA + angle CQA <-- substitution

= angle BQC <-- angle addition postulate

[end of proof]