What is the measure of side BC?

If BD=5x and CD=4x+3, what is the measure of BC?

Prove whether or not DB is the perpendicular bisector of AC. If the proof is impossible what would you need to know to make a proof.

Prove whether or not DC is the perpendicular bisector of AB. If the proof is impossible what would you need to know to make a proof.

If[math]\angle[/math]ABD is 50[math]^\circ[/math], what is the measure of [math]\angle[/math]ABC?

What is the measure of segment CD?

Prove whether or not AE is the angle bisector of [math]\angle[/math]BAC. If the proof is impossible what would you need to know to make a proof.

Prove whether or not BD is the angle bisector of [math]\angle[/math]ABC. If the proof is impossible what would you need to know to make a proof.

You are building some shelves in your room. You know that triangles are the strongest shape. So, you decide some wire bracing would be helpful. The end result looks like the diagram above. Unfortunately, there's someone in your life that keeps bumping the shelf and knocking things over. You have a rigid piece of PVC that is 4 units long. Is there a way to us it to brace both sides of the shelf with the PVC? If so, how? If not, why not? (CF and CD are congruent to BA. All angles at points B and C are perpendicular.)