Look for congruent sides of triangles △ MLP and △ QLN.
See solution.
Practice makes perfect
We are asked to show that the two shaded overlapping triangles are congruent.
Let's mark the given congruent angles and segments on the diagram.
We can see that there are two pairs of congruent angles in triangles △ MLP and △ QLN.
Angle ∠ 3 of triangle △ M L P is congruent to angle ∠ 2 of triangle △ Q L N.
Angle ∠ M of triangle △ M L P is congruent to angle ∠ Q of triangle △ Q L N.
Let's see what can we say about the included segments M P and N Q.
It is given that MN and PQ are congruent, so by definition they have the same length.
MN=PQ
We can add NP to both sides.
MN+NP=NP+PQ
According to the Segment Addition Postulate, MN+NP=MP and NP+PQ=NQ.
This means that MP=NQ, so by definition MP and NQ are congruent segments.
M P≅N Q
Let's summarize now what we know about triangles △ M L P and △ Q L N.
The two triangles have two pairs of congruent angles and the included sides are also congruent.
According to the Angle-Side-Angle (ASA) Congruence Postulate, this means that the two triangles are congruent.
We can summarize the process above in a flow proof.
