Question

Ray AK bisects ∠BAC, point M ∈ AK so that m∠AMB = m∠AMC. Prove that BK = CK.

Answer 1

Answer:

Given information: AK bisects ∠BAC, point M ∈ AK so that m∠AMB = m∠AMC.

Prove : BK = CK.

Proof:

In triangle ABM and ACM,

$\angle AMB\cong \angle AMC$                  (Given)

$AM=AM$                   (Reflection property)

$\angle MAB\cong \angle MAC$                  (Definition of angle bisector)

By SAS postulate,

$\triangle AMB\cong \triangle AMC$

$AB\cong AC$                             (CPCTC)

$AB=AC$

It means triangle ABC is an isosceles triangle.

The angle bisector of an isosceles triangle divides the non equal side in two equal parts.

$BK=CK$

Hence proved.