Proving congruence using two column proof, question is in attached picture

Question

Answer 1

So some of this depends on which theorems and reasoning your particular geometry class uses/permits. In addition, your class may use different notations (such as congruent symbols) that I can’t type here, so please be wary of that. However, the basic idea is as follows:

KE = KJ (Proof: BD bisects KJ at E)

1)     Similarly, BE=DE (Proof: KJ bisects BD at E)

Angle BEI = Angle DEK (Proof: Opposite angles are equal)

Triangle DEK is congruent to triangle BEI (Proof: Side-Angle-Side Congruence)

2)     Angle KDE = Angle EBJ (Proof: Congruent parts of congruent triangles are congruent)

Angle CEK=Angle JEA (Proof: Assuming CEA forms one line, then opposite angels are equal. You may first have to show that CEK and JEA are opposite angels)

3)     Angle BEA = Angle CED (Proof: Angle BEJ= Angle DEK, and Angle JEA = Angle CEK. Since BEA is composed of BEJ and JEA while CED is composed of DEK and CEK, the two must be equal since the angles that form them are equal).

Triangle CED and Triangle BEA are congruent (Proof: Angle-Side-Angle Congruence using 1, 2, and 3)

Thus, CE = AE (Proof: Congruent Parts of Congruent Triangles are congruent)

The way to see this is to realize that opposite angles and bisecting lines help set up congruent triangles. You can use this fact to see that CED and BEA are also congruent, which gives you your solution.