Congruent triangles have congruent corresponding sides and angles.
A perpendicular bisector, median, and an altitude.
Practice makes perfect
We are told that △ JLM is congruent to △ KLM. Corresponding sides of congruent triangles are congruent. In our case, side JM in △ JLM corresponds to MK in △ KLM. Thus, they are congruent segments and have the same measures. This allows us to conclude that M is a midpoint of JK.
Now, let's review that a median of a triangle is a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side. LM fully satisfies this definition, so it is a median of △ JLK.
We can also use the fact that corresponding angles of congruent triangles are congruent. In △ LMJ, ∠ JML is corresponding to ∠ KML in △ LMK. Hence, these are congruent angles, which means that their measures are the same.
m∠ JML=m∠ KML
Moreover, these angles are supplementary and the sum of their measures is 180^(∘).
m∠ JML+ m∠ KML=180^(∘)
Let's substitute m∠ JML for m∠ KML in the above equality and calculate the measure of ∠ JML.
Therefore, ∠ JML is a right angle and so is ∠ KML. This means that LM is perpendicular to JK. We can conclude that LM is also an altitude of △ JLK.
So far we know that LM is perpendicular to JK and bisects JK into two congruent segments, JM and MK. Therefore, LM is a perpendicular bisector of JK.
The final answer is that LM is all three things — a median, an altitude, and a perpendicular bisector of △ JLK.
