When a line bisects an angle of a triangle, it cuts the opposite side into two pieces in the same ratio as the two sides next to the angle. With AD bisecting angle A, AB over AC equals BD over DC: each side goes with the piece of BC it touches.
Not quite the one you need? Same words, different skill:
Pairing each side with the piece across from it. AB touches BD, so they go together; pairing AB with DC turns the proportion upside down and gives the wrong length.
Also called angle bisector theorem, triangle angle bisector theorem or angle bisector proportion.
What you can put on this worksheet
The angle bisector theorem
AD is the bisector of angle BAC. Find x.— triangle a b c with a d bisecting angle a and meeting b c at d, labeled A, B, C, D, 20, 8, 10, x→ 4
The whole opposite side given
The bisector of ∠Ameets BCat D,and BC=40.Find x.— triangle a b c with a d bisecting angle a and meeting b c at d, labeled A, B, C, D, 30, x, 24→ 20
Questions about these worksheets
Yes. Every triangle angle bisector theorem sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.
Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.
Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.
Geometry is usually taken around 9th or 10th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.
Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.
That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the triangle angle bisector theorem questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.
Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.
You pick from 2: the angle bisector theorem and the whole opposite side given. Tick as many as you want and set how many of each, or let it spread them evenly.
Pairing each side with the piece across from it. AB touches BD, so they go together; pairing AB with DC turns the proportion upside down and gives the wrong length.