If point C is the midpoint of both BD and AE, what additional information can you deduct from the diagram to be able to state the triangle ABC= triangle EDC by the SAS congruence theorem?

Question

A. line BC = line DC
B. Line AB = line ED
C. <ACB = <ECD
D. <A =<E

Answer 1

To apply the SAS (Side-Angle-Side) congruence theorem to prove that triangle ABC is congruent to triangle EDC, we need to establish two pairs of corresponding sides that are equal, along with the included angle between those sides.

Given that C is the midpoint of both BD and AE, we have:

• BC = CD (since C is the midpoint of BD)
• AC = EC (since C is the midpoint of AE)

Now, for the SAS theorem, we need one angle to be congruent as well.

The options given are:

A. line BC = line DC
B. Line AB = line ED
C. <ACB = <ECD
D. <A = <E

Since we already have one pair of equal sides (BC = CD) and need another side and the included angle, the statement about the angles would be relevant for SAS.

Between the options, C: <ACB = <ECD provides that relationship, as we need that angle to show the congruence of triangles ABC and EDC.

Therefore, the additional information you can deduct from the diagram to be able to state that triangle ABC is congruent to triangle EDC by the SAS congruence theorem is:

C. <ACB = <ECD