For a two column proof, we want to start with the given information. From there, we will use various definitions, postulates, and theorems to fill in the rest.
Our two sets of given information are that Plane M bisects Line AB and that Line PA is congruent to Line PB.
We know from the definition of a bisector that it splits a line in two equal parts. Therefore, Line AO must be congruent to Line BO.
Now, we have two sides of a triangle that we have proved to be congruent to each other. From the image given in the original problem, we see that both triangles share Line OP. Line OP is congruent to Line OP through the reflexive property.
We now have proven that all three sides of the one triangle are congruent to the corresponding sides on the other triangle. Therefore, the triangles are congruent through the SSS theorem.
