If point C is the midpoint of both BD and AE what additional information would you need to be able to state the triangle ABC is congruent to triangle EDC by the SAS congruence theorem?

Question

Line BC is congruent to line DC
Line AB is congruent to line ED
Angle ACB is congruent to angle ECD
Angle A is congruent to angle E

Answer 1

To prove that triangle ABC is congruent to triangle EDC by the SAS (Side-Angle-Side) congruence theorem, we need to establish two pairs of congruent sides and the included angle between them.

According to the information provided:

1. Line BC is congruent to line DC (this gives us one pair of congruent sides).
2. Line AB is congruent to line ED (this gives us the second pair of congruent sides).
3. Angle ACB is congruent to angle ECD (this is the included angle between the two pairs of congruent sides).

Since you mentioned that point C is the midpoint of both BD and AE, we can derive that:

• Line BC = Line DC (from point C being the midpoint of BD).
• Line AB = Line ED (from point C being the midpoint of AE).

Thus, to use the SAS theorem, we would only need to confirm that the angle between line AB and line AC is congruent to the angle between line ED and line EC, specifically:

Angle ACB is congruent to angle ECD.

If we have these conditions met, we can state that triangle ABC is congruent to triangle EDC by the SAS theorem.

Therefore, the additional information you would need is:

Angle ACB is congruent to angle ECD.