A perpendicular bisector of a segment is a line perpendicular to the segment that goes through its midpoint. Therefore, we know that YG divides DF in two equal halves and cuts it at a right angle. Let's add this information to the diagram.
Notice that △ DEY and △ FEY share a side YE. By the Reflexive Property of Congruence, we know that YE is congruent between the two triangles. Let's add this information to the diagram.
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
Note that ∠ DEY is the included angle of DE and EY, and ∠ FEY is the included angle of FE and EY. Therefore, we can use the Side-Angle-Side Congruence Theorem to show that △ DEY ≅ △ FEY. Let's show this as a two-column proof.
Statement
Reason
1.
YG is the perpendicular bisector of DF
1.
Given
2.
&∠ YEF is a right angle &∠ YED is a right angle &DE ≅ FE
