We asked to show that the segments which connect the midpoint of a side of a rectangle with the opposite vertices are congruent. Let's focus on the triangles, which are formed in part because of the segments in question, shaded in the diagram.
Focusing on congruent relationships, let's summarize what we know about these triangles.
Congruence
Justification
BC≅DC
Given
∠ B≅∠ D
The angles of a rectangle are right angles, and all right angles are congruent.
AB≅ED
A rectangle is a parallelogram, and opposite sides of a parallelogram are congruent (Theorem 6.3).
We can see that the two sides and the included angle of triangle △ ABC are congruent to the two sides and the included angle of triangle △ EDC. Given those relationships, according to the Side-Angle-Side (SAS) Congruence Postulate, these two triangles are congruent.
△ ABC≅△ EDC
Segments AC and EC are corresponding sides of these congruent triangles, so they are congruent.
AC≅EC
We can summarize the process above in a flow proof.
