It is given that angle ABR and angle ACR are right angles, line segment AB is congruent to line segment BC and line segment BC is congruent to line segment AC Since they contain right angles, triangle ABR and triangle ACR are right triangles. The right triangles share hypotenuse line segment AR , and reflexive property justifies that line segment AR is congruent to line segment AR Since line segment AB is congruent to line segment BC and line segment BC is congruent to line segment AC , the transitive property justifies line segment AB is congruent to line segment AC Now, the hypotenuse and leg of right triangle ABR is congruent to the hypotenuse and the leg of right triangle ACR , so triangle ABR is congruent to triangle ACR by the HL congruence postulate. Therefore, ________ by CPCTC, and line segment AR bisects angle BAC by the definition of bisector.
Answer
A
angle APB is congruent to angle APC
B
angle BAR is congruent to angle CAR
C
angle APC is congruent to angle BPR
D
angle BPR is congruent to angle CPR
Explanation
The triangles are congruent with the correspondence A ↔ A, B ↔ C, and R ↔ R. By CPCTC, the corresponding angles at A are congruent: ∠BAR ≅ ∠CAR. This is choice B, and it shows that AR bisects ∠BAC.
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