Question

Given:

CM bisects AB
AC = BC

Based on the given information and the algebraic and geometric properties presented or proven thus far, choose the congruence theorem that could be used to prove the triangles congruent. If it is not possible to prove the triangles are congruent, choose "not possible".

SSS
SAS
ASA
not possible

Answer 1

Answer: The correct option is 1, i.e., SSS.

Step-by-step explanation:

It is given that in triangle ABC the line CM bisects AB. The side AC=BC. In triangle ACM and BCM,

AM = BM (CM bisects AB)

CM=CM (common side)

AC= BC (given)

Since three sides of triangles are equal.

According to SSS rule of congruence theorem, two triangles are congruent if all the corresponding sides of the triangles are equal.

So by SSS rule of congruence theorem, we can conclude that the triangles ACM and BCM are congruent.

The correct option is first, i.e., SSS rule.

Answer 2

We can use the SSS congruence theorem to prove that the two triangles in the attached figure are congruent. The SSS or side-side-side theorem states that each side in the first triangle must have the same measurement or must be congruent on each of the opposite side of another triangle. In this problem, for the first triangle, we have sides AC, CM, AM while in the second triangle we have sides BC, CM, and BM. By SSS congruent theorem, we have the congruent side as below:
AC = BC
CM = CM
AM = BM

The answer is SSS theorem.