Complete the paragraph proof.

Question

Given: and are right angles

Line segment A B is-congruent-to line segment B C Line segment B C is-congruent-to line segment A C

Prove: Line A R bisects Angle B A C

Triangles A B R and R C A share side R A. A line is drawn from point B to point C and intersects side A R at point P.

It is given that and are right angles, and . Since they contain right angles, ΔABR and ΔACR are right triangles. The right triangles share hypotenuse , and reflexive property justifies that . Since and , the transitive property justifies . Now, the hypotenuse and leg of right ΔABR is congruent to the hypotenuse and the leg of right ΔACR, so by the HL congruence postulate. Therefore, ________ by CPCTC, and bisects by the definition of bisector.

Answer 1

Given: ∠ABR and ∠ACR are right angles

Line segment AB is congruent to line segment BC. Line segment BC is congruent to line segment AC.

Prove: Line AR bisects Angle BAC.

Triangles ABR and ACR share side AR. A line is drawn from point B to point C and intersects side AR at point P.

It is given that ∠ABR and ∠ACR are right angles, AB ≅ BC, and BC ≅ AC. Since they contain right angles, ΔABR and ΔACR are right triangles. The right triangles share hypotenuse AR, and the reflexive property justifies that AR ≅ AR. Since AB ≅ AC, the transitive property justifies that AB ≅ AC. Now, the hypotenuse and leg of right ΔABR is congruent to the hypotenuse and the leg of right ΔACR, so ΔABR ≅ ΔACR by the HL congruence postulate. Therefore, ∠ABR ≅ ∠ACR by CPCTC, and AR bisects ∠BAC by the definition of bisector.