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Understanding congruent triangles is fundamental in proving various geometric propositions. Two triangles are considered congruent if they have exactly the same three sides and three angles. This congruency implies that all corresponding pairs of sides and angles are equal. In more simple terms, if you could pick up one triangle and place it on top of the other, they would match perfectly.

In the context of our exercise, where we look at the triangle ABC with altitudes AD and BE that are equal, we have two right triangles, triangle ABD and triangle ABE. These two triangles share a hypotenuse (AB) and have equal altitudes (AD and BE), which serve as one of their legs. By assessing their properties, we can compare and determine the equality of other triangle elements, like the remaining sides and angles, hence concluding congruence between these two right triangles.

The ASA Congruence Theorem is a powerful tool used to prove that two triangles are congruent. ASA stands for Angle-Side-Angle, meaning that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

In our case, for triangles ABD and ABE, we know that angle BAD equals angle BAE (both are 90 degrees), and we have been given that AD equals BE. The side AB is common to both triangles. Following the logic of the ASA Congruence Theorem, this is sufficient evidence to state that triangle ABD is congruent to triangle ABE. It allows us to infer that all their corresponding angles and sides are identical, facilitating the next steps in the proof of the isosceles nature of the main triangle ABC.

The concept of triangle altitudes is another crucial element in understanding triangles. An altitude of a triangle is a segment from a vertex to the line containing the opposite side, and it's perpendicular to that side. Every triangle has three altitudes, and they can be inside or outside the triangle, depending on the type of triangle.

It's interesting to note that in an isosceles triangle, the altitudes from the vertices at the base are equal, just as in our exercise with triangle ABC having equal altitudes AD and BE. This property arises from the symmetry of the isosceles triangle. When these altitudes are equal, they can often be the key to unlocking the triangle's other properties, such as the lengths of sides or angles, as seen through the congruence of triangles ABD and ABE.