Triangle Congruence Problem: Proving DCE ≅ ABE with 50° Angles

Determine whether the triangles DCE and ABE congruent?

If so, according to which congruence theorem?

Congruent triangles are triangles that are identical in size, meaning that if we place one on top of the other, they will match exactly.

In order to prove that a pair of triangles are congruent, we need to prove that they satisfy one of these three conditions:

1. SSS - Three sides of both triangles are equal in length.

2. SAS - Two sides are equal between the two triangles, and the angle between them is equal.

3. ASA - Two angles in both triangles are equal, and the side between them is equal.

If we take an initial look at the drawing, we can already observe that there is one equal side between the two triangles, as they are both marked in blue,

We don't have information regarding the other sides, thus we can rule out the first two conditions,

And now we'll focus on the last condition - angle, side, angle.

We can observe that angle D equals angle A, both equal to 50 degrees,

Let's proceed to the angles E.

At first glance, one might think that there's no way to know if these angles are equal, however if we look at how the triangles are positioned,
We can see that these angles are actually corresponding angles, and corresponding angles are of course equal.

Therefore - if the angle, side, and second angle are equal, we can prove that the triangles are equal using the ASA condition