Question

How would you prove the triangles congruent?

First, prove that /AB is congruent to /BC because B is equidistant from A and C. Then use SSS to prove the triangles are congruent.

First, prove that ∠E is congruent to ∠D because congruent parts of congruent triangles are congruent. Then use SAS to prove the triangles are congruent.

First, prove that ∠ABE is congruent to ∠CBD because vertical angles are congruent. Then use SSA to prove the triangles are congruent.

There is not enough information to prove that the triangles are congruent.

Answer 1

The answer is:
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[C]: First, prove that ∠ABE is congruent to ∠CBD because vertical angles are congruent. Then use SSA to prove the triangles are congruent.
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Answer 2

There is not enough information to prove that the triangles are congruent.

Explanation:

Two figures are congruent if they have the same shape and size, so triangle ABE and triangle CBD are congruent if they have exactly the same three angles and exactly the same three sides.

The first choice says that AB ≅ BC because B is equidistant from A and C. Unless we know that AE and DC are vertical or parallel, we can't say that B is equidistant from A and C.

The second choice says that ∠E ≅ ∠D because congruent parts of congruent triangles are congruent. Since we are proving that triangle ABE and triangle CBD are congruent, we can't say that two triangles are congruent because they are congruent, it does not make sense at all.

Finally, the third choice says that ∠ABE is congruent to ∠CBD because vertical angles are congruent, which is true, but SSA is not a method to prove congruence. If two triangles have two congruent sides and a congruent nonincluded angle, then triangles are not necessarily congruent. The only methods to prove congruence are: SSS, SAS, ASA, AAS and HL.