Angle bisectors worksheet

A bisector cuts an angle into two equal halves. So either half is the whole angle divided by two, the whole is either half doubled, and when the two halves are given as expressions they are equal to each other.

Not quite the one you need? Same words, different skill:

The Triangle Angle Bisector Theorem (Geometry). A bisector splits the far side in the ratio of the near sides.

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Sigma Prep · sigmaprep.io/worksheets Name Angle Bisectors Date Period Find each measure. Find angles with a bisector $\vec{B D}$ bisects $∠ A B C$ , and $m ∠ A B C = 168^{\circ}$ . Find $x$ . $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 27)}^{\circ}$ and $m ∠ A B C = {(3 x + 47)}^{\circ}$ . Find $m ∠ A B C$ . $\vec{B D}$ bisects $∠ A B C$ . Find $x$ . $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 88^{\circ}$ . Find $m ∠ A B D$ . $\vec{B D}$ bisects $∠ A B C$ . What is $m ∠ A B C$ ? $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(5 x - 7)}^{\circ}$ and $m ∠ A B C = {(6 x + 34)}^{\circ}$ . Find $m ∠ A B C$ . $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 100^{\circ}$ . Find $m ∠ D B C$ . $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 16)}^{\circ}$ and $m ∠ A B C = {(5 x - 4)}^{\circ}$ . Find $m ∠ A B C$ . $\vec{B D}$ bisects $∠ A B C$ . Solve for $x$ . $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 58^{\circ}$ . Find $m ∠ A B D$ . $\vec{B D}$ bisects $∠ A B C$ , and $m ∠ A B C = {(5 x - 34)}^{\circ}$ . Find $x$ . $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(6 x + 31)}^{\circ}$ and $m ∠ A B C = {(x + 106)}^{\circ}$ . Find $m ∠ A B C$ . $\vec{B D}$ cuts $∠ A B C$ into two equal angles. Find $m ∠ A B C$ . $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(5 x + 7)}^{\circ}$ and $m ∠ A B C = {(5 x + 54)}^{\circ}$ . Find $m ∠ A B C$ . $\vec{B D}$ bisects $∠ A B C$ . Find $m ∠ A B C$ . $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 66^{\circ}$ . Find $m ∠ A B D$ . $∠ A B C$ is cut in half by $\vec{B D}$ . Find the value of $x$ . $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 48)}^{\circ}$ and $m ∠ A B C = {(x + 100)}^{\circ}$ . Find $m ∠ A B C$ . $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 140^{\circ}$ . Find $m ∠ D B C$ . $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(2 x + 9)}^{\circ}$ and $m ∠ A B C = {(6 x - 10)}^{\circ}$ . Find $m ∠ A B C$ . Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Angle Bisectors

Date Period

Find each measure.

Find angles with a bisector
$\vec{B D}$ bisects $∠ A B C$ , and $m ∠ A B C = 168^{\circ}$ . Find $x$ .
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 27)}^{\circ}$ and $m ∠ A B C = {(3 x + 47)}^{\circ}$ . Find $m ∠ A B C$ .
$\vec{B D}$ bisects $∠ A B C$ . Find $x$ .
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 88^{\circ}$ . Find $m ∠ A B D$ .
$\vec{B D}$ bisects $∠ A B C$ . What is $m ∠ A B C$ ?
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(5 x - 7)}^{\circ}$ and $m ∠ A B C = {(6 x + 34)}^{\circ}$ . Find $m ∠ A B C$ .
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 100^{\circ}$ . Find $m ∠ D B C$ .
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 16)}^{\circ}$ and $m ∠ A B C = {(5 x - 4)}^{\circ}$ . Find $m ∠ A B C$ .
$\vec{B D}$ bisects $∠ A B C$ . Solve for $x$ .
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 58^{\circ}$ . Find $m ∠ A B D$ .
$\vec{B D}$ bisects $∠ A B C$ , and $m ∠ A B C = {(5 x - 34)}^{\circ}$ . Find $x$ .
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(6 x + 31)}^{\circ}$ and $m ∠ A B C = {(x + 106)}^{\circ}$ . Find $m ∠ A B C$ .
$\vec{B D}$ cuts $∠ A B C$ into two equal angles. Find $m ∠ A B C$ .
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(5 x + 7)}^{\circ}$ and $m ∠ A B C = {(5 x + 54)}^{\circ}$ . Find $m ∠ A B C$ .
$\vec{B D}$ bisects $∠ A B C$ . Find $m ∠ A B C$ .
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 66^{\circ}$ . Find $m ∠ A B D$ .
$∠ A B C$ is cut in half by $\vec{B D}$ . Find the value of $x$ .
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 48)}^{\circ}$ and $m ∠ A B C = {(x + 100)}^{\circ}$ . Find $m ∠ A B C$ .
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 140^{\circ}$ . Find $m ∠ D B C$ .
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(2 x + 9)}^{\circ}$ and $m ∠ A B C = {(6 x - 10)}^{\circ}$ . Find $m ∠ A B C$ .



Sigma Prep · sigmaprep.io/worksheets Name Angle BisectorsAnswer keyVersion 1 Date Period Find each measure. Find angles with a bisector $\vec{B D}$ bisects $∠ A B C$ , and $m ∠ A B C = 168^{\circ}$ . Find $x$ . $x = 12$ $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 27)}^{\circ}$ and $m ∠ A B C = {(3 x + 47)}^{\circ}$ . Find $m ∠ A B C$ . $68^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ . Find $x$ . $x = 21$ $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 88^{\circ}$ . Find $m ∠ A B D$ . $44^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ . What is $m ∠ A B C$ ? $80^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(5 x - 7)}^{\circ}$ and $m ∠ A B C = {(6 x + 34)}^{\circ}$ . Find $m ∠ A B C$ . $106^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 100^{\circ}$ . Find $m ∠ D B C$ . $50^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 16)}^{\circ}$ and $m ∠ A B C = {(5 x - 4)}^{\circ}$ . Find $m ∠ A B C$ . $56^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ . Solve for $x$ . $x = 22$ $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 58^{\circ}$ . Find $m ∠ A B D$ . $29^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ , and $m ∠ A B C = {(5 x - 34)}^{\circ}$ . Find $x$ . $x = 18$ $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(6 x + 31)}^{\circ}$ and $m ∠ A B C = {(x + 106)}^{\circ}$ . Find $m ∠ A B C$ . $110^{\circ}$ $\vec{B D}$ cuts $∠ A B C$ into two equal angles. Find $m ∠ A B C$ . $100^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(5 x + 7)}^{\circ}$ and $m ∠ A B C = {(5 x + 54)}^{\circ}$ . Find $m ∠ A B C$ . $94^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ . Find $m ∠ A B C$ . $62^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 66^{\circ}$ . Find $m ∠ A B D$ . $33^{\circ}$ $∠ A B C$ is cut in half by $\vec{B D}$ . Find the value of $x$ . $x = 16$ $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 48)}^{\circ}$ and $m ∠ A B C = {(x + 100)}^{\circ}$ . Find $m ∠ A B C$ . $104^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 140^{\circ}$ . Find $m ∠ D B C$ . $70^{\circ}$ $\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(2 x + 9)}^{\circ}$ and $m ∠ A B C = {(6 x - 10)}^{\circ}$ . Find $m ∠ A B C$ . $74^{\circ}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Angle BisectorsAnswer keyVersion 1

Date Period

Find each measure.

Find angles with a bisector
$\vec{B D}$ bisects $∠ A B C$ , and $m ∠ A B C = 168^{\circ}$ . Find $x$ . $x = 12$
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 27)}^{\circ}$ and $m ∠ A B C = {(3 x + 47)}^{\circ}$ . Find $m ∠ A B C$ . $68^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ . Find $x$ . $x = 21$
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 88^{\circ}$ . Find $m ∠ A B D$ . $44^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ . What is $m ∠ A B C$ ? $80^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(5 x - 7)}^{\circ}$ and $m ∠ A B C = {(6 x + 34)}^{\circ}$ . Find $m ∠ A B C$ . $106^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 100^{\circ}$ . Find $m ∠ D B C$ . $50^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 16)}^{\circ}$ and $m ∠ A B C = {(5 x - 4)}^{\circ}$ . Find $m ∠ A B C$ . $56^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ . Solve for $x$ . $x = 22$
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 58^{\circ}$ . Find $m ∠ A B D$ . $29^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ , and $m ∠ A B C = {(5 x - 34)}^{\circ}$ . Find $x$ . $x = 18$
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(6 x + 31)}^{\circ}$ and $m ∠ A B C = {(x + 106)}^{\circ}$ . Find $m ∠ A B C$ . $110^{\circ}$
$\vec{B D}$ cuts $∠ A B C$ into two equal angles. Find $m ∠ A B C$ . $100^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(5 x + 7)}^{\circ}$ and $m ∠ A B C = {(5 x + 54)}^{\circ}$ . Find $m ∠ A B C$ . $94^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ . Find $m ∠ A B C$ . $62^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 66^{\circ}$ . Find $m ∠ A B D$ . $33^{\circ}$
$∠ A B C$ is cut in half by $\vec{B D}$ . Find the value of $x$ . $x = 16$
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(x + 48)}^{\circ}$ and $m ∠ A B C = {(x + 100)}^{\circ}$ . Find $m ∠ A B C$ . $104^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 140^{\circ}$ . Find $m ∠ D B C$ . $70^{\circ}$
$\vec{B D}$ bisects $∠ A B C$ . $m ∠ A B D = {(2 x + 9)}^{\circ}$ and $m ∠ A B C = {(6 x - 10)}^{\circ}$ . Find $m ∠ A B C$ . $74^{\circ}$

How to do these

Ray $B D$ bisects $∠ A B C$ . If $∠ A B D = 42^{\circ}$ , find $∠ A B C$

ABD is half of the whole angle.
So the whole is twice 42.

The answer is $84^{\circ}$ .

Where students go wrong

Halving when you should double. If you are given one of the halves, the whole angle is bigger, not smaller. Check which one the question names before dividing.

Also called bisecting an angle or angle bisector.

What you can put on this worksheet

Angle bisectors: $∠ A B C$ is cut in half by $\vec{B D}$ . Find the value of $x$ . — angle a b c with ray b d drawn down the middle, labeled B, A, D, C → $x = 3$
Bisectors: whole to half, and both in x: $\vec{B D}$ bisects $∠ A B C$ and $m ∠ A B C = 100^{\circ}$ . Find $m ∠ A B D$ . — angle a b c with ray b d between its sides, labeled A, D, C, B → $50^{\circ}$

Questions about these worksheets

Yes. Every angle bisectors 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 angle bisectors 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: angle bisectors and bisectors: whole to half, and both in x. Tick as many as you want and set how many of each, or let it spread them evenly.

Halving when you should double. If you are given one of the halves, the whole angle is bigger, not smaller. Check which one the question names before dividing.

Angle bisectors worksheet

Find angles with a bisector

Find angles with a bisector

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

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