Question

Can someone help me with this two column proofs? -this one is confusing to me

Answer 1

Perpendicular bisectors have a particular property: if AB is a perpendicular bisector of CD, then every point lying on AB has the same distance from C and D.

In your case, we have that every point lying on AC has the same distance from B and D.

So, in particular, we have EB=ED, because E lies on AC.

Moreover, since AC is a perpendicular bisector, it is the height of the triangle (if we choose BD as base), and it bisects BD: this means that the triangle is isosceles, so AD=AB.

This means that triangles ABE and ADE have:

• AD=AB because ABD is isosceles
• EB=ED because AC is the perpendicular bisector of BD
• AE in common

So, their sides are all equal, and thus they are congruent.