Question

Complete the paragraph proof. Given: ∠ABR and ∠ACR are right angles AB ≅ BC BC ≅ AC Prove: bisects ∠BAC

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

Answer 1

The answer is <BAR ≅<CAR :)

Answer 2

∠BAR=∠CAR [∴ΔABR≅ΔACR]

Explanation:

In ΔABR and ΔACR

AB=AC=X [ ∴AB≅BC, and BC≅AC, So AB≅AC]

∠ABR = ∠ACR [ each being 90°]

AR is common.

ΔABR ≅ ΔACR [ RHS]

RHS means if in two right triangles hypotenuse and one side of a triangle is equal to hypotenuse and other side then the two triangles are congruent.

So ,∠BAC is bisected.

i.e, ∠BAR=∠CAR [ CPCT ]

Hence proved.