∠ABE=∠CBE

∠ BAC = 90, and AD is the height on the side of BC.

∠ADB=90

∠ADB+∠PBD=∠BAC+∠ABE

The sum of the internal angles of a triangle is 180, so the other internal angles are equal.

∠BPD=∠AEP

Equal vertex angle

∠BPD =∠ ape

∠ ape =∠AEP

AE=AP

2.

Let EF⊥CD pass through e and give the CD to F.

Accord to that properties of the angular bisector

The distance between points on the bisector of an angle is equal to both sides of the angle.

AE=EF= 1

The right-angled side facing 30 is half of the hypotenuse.

EF= 1/2EC

EC=2

AC=AE+EC=2+ 1=3