Trapezoid Angle Bisector Problem: Find AB

A classic geometry problem, worked out step by step by Dr. E — with the full reasoning behind every move, a video walkthrough, and a free practice worksheet.

📐 Geometry 🎬 Video solution 📄 Free worksheet
The Problem
A D C B P 4 9
Figure not drawn to scale

In trapezoid ABCD, AD ∥ BC. The bisectors of ∠A and ∠B intersect at point P on side DC. Given BC = 9 and AD = 4, find AB.

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Video solution coming this week. Dr. E walks through this exact problem on the whiteboard, step by step.

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Step-by-Step Solution
1
Extend AP until it meets line BC at point E.
Because AD ∥ BC with transversal AE, the alternate interior angles are equal: ∠DAE = ∠AEB. But AP bisects ∠A, so ∠DAE = ∠EAB. Putting these together: ∠EAB = ∠AEB.
Two equal base angles → triangle ABE is isosceles, with AB = BE.
2
BP is the bisector from the apex of isosceles △ABE — so it is also a median.
In an isosceles triangle, the angle bisector drawn from the apex (here B, since BA = BE) is simultaneously the median and the altitude. P lies on that bisector and on AE, so P is the midpoint of AE: AP = PE.
3
Triangles ADP and ECP are congruent (ASA).
AP = PE (step 2), ∠APD = ∠EPC (vertical angles), and ∠DAP = ∠CEP (alternate interior angles, AD ∥ EC). Therefore △ADP ≅ △ECP, which gives CE = AD = 4.
Bonus: the congruence also gives DP = PC — P is the midpoint of DC.
4
Add the segments along line BC.
AB = BE = BC + CE = 9 + 4 = 13.
Answer AB = 13 In general: whenever the bisectors of the two angles on one leg of a trapezoid meet on the other leg, that first leg equals the sum of the parallel sides — here AB = AD + BC.

💡 Why this works — the picture to remember

The whole solution is one idea: a bisector plus parallel lines creates an isosceles triangle. The moment you extend AP to line BC, the alternate-interior angles "copy" half of ∠A down to E, forcing AB = BE. Everything after that is bookkeeping.

Two elegant facts hide inside this configuration: P is always the midpoint of DC, and since ∠A + ∠B = 180° (co-interior angles), the half-angles add to 90° — so ∠APB is always a right angle. Both make great extension questions for students.

📄 Practice this skill — free worksheet Angle bisectors & perpendicular bisectors PDF — more problems just like this one. 📐 All geometry video lessons 7 units from foundations to circles — free, in English and Filipino. 🎬 Numberbender on YouTube 700+ free math lessons by Dr. E. Subscribe for new solved problems.
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