KS3 · Maths

Bisecting an angle

You can cut an angle perfectly in half without knowing its size. No protractor, no numbers: just compasses, a ruler, and a shape hiding inside every angle.

Maths · Geometry

Can you break the rhombus and still split the angle in half?
124°62°62°5.0 cm5.0 cm5.0 cm5.0 cmOPQR

Whole angle POQ 124°. Half: POR 62°. Half: ROQ 62°. OP 5.0 cm. OQ 5.0 cm. PR 5.0 cm. QR 5.0 cm. Relationship: When OP = OQ = PR = QR, OPRQ is a rhombus, and its diagonal OR splits angle POQ into two equal halves.

Whole angle POQ124°Half: POR62°Half: ROQ62°OP5.0 cmOQ5.0 cmPR5.0 cmQR5.0 cmdrag Q, then slide R

When OP = OQ = PR = QR, OPRQ is a rhombus, and its diagonal OR splits angle POQ into two equal halves.

The compasses are set to 5 cm. Q can only move along the first arc (centre O, radius 5 cm), so dragging Q changes the angle. R can only move along the arc from P, same width. Change the angle, then slide R until QR is back to 5.0 cm, like the other three sides, and watch the two halves.

Watch out: The two halves only match when QR matches the other three sides. (The arcs from P and Q also meet at O itself, so aim for the other crossing.) In a real construction you never draw PR and QR: the arcs fix R, and the line from R through O is all you need.

Why the arcs work

?

Reason it through

Why does the line from the arc crossing through the vertex split the angle exactly in half?

Link 1 of 4

First link · your turn

The first arc is drawn with the compass point on the vertex O. It crosses the arms at P and Q. What do you know about OP and OQ?

2
Locked — reveal the link above first
3
Locked — reveal the link above first
4
Locked — reveal the link above first

Maths · Constructions

Put the construction in order

The order you do each step in when you bisect an angle

1 · Do this first5 · Do this last

  1. Draw a straight line from where the two arcs cross, through the vertex.

  2. Put the compass point on the vertex and draw an arc that crosses both arms.

  3. Leave all the arcs on the page.

  4. Set the compasses to a sensible width, no longer than the shorter arm.

  5. Keep the same width. Put the compass point on each place where the first arc crossed an arm, and draw two arcs that cross.

Maths · Constructions

Which of these is bisecting by construction?

Your teacher draws a 70° angle and asks you to bisect it by construction.

Which is closest to what you'd do?
How sure are you?

Maths · Checking a construction

Where did Sam's construction go wrong?

Bisect angle ABC, which is 84°. The arms BA and BC are 6 cm and 8 cm long.

Sam's construction — which line goes wrong?

Maths · Reflex angles

Bisect a reflex angle

Two arms meet at O and make a reflex angle of 250°. Bisect the reflex angle.

  1. The 250° is the angle round the outside of the corner. On the other side, the same two arms make a smaller angle.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

  • What it means to bisect an angle, and why measuring and halving isn't a construction.
  • How to bisect an angle with a ruler and compasses, in the right order.
  • Why the method works: equal arcs build a rhombus, and a rhombus's diagonal halves its angle.
  • How to check a bisector with a protractor, and how to bisect a reflex angle.

The big picture

To bisect an angle is to split it into two equal angles. With one compass width, draw an arc from the vertex across both arms, draw two arcs from where it crosses them, and draw a line from where those arcs meet through the vertex. The equal arcs build a rhombus, and a rhombus's diagonal always cuts its angle in half.

Key points

1To bisect an angle means to split it into two equal angles. The line that does it is the angle bisector.
2A construction uses a ruler and compasses only. Measuring with a protractor and halving isn't a construction; the protractor is for checking afterwards.
3Method: arc from the vertex across both arms; same width, arcs from those two points so they cross; straight line from the crossing through the vertex. Keep one width throughout, and don't rub out construction lines.
4The compass width can be anything up to the length of the shorter arm, and arcs are enough: full circles aren't needed.
5Why it works: the equal arcs make a rhombus, and a rhombus's diagonals bisect its angles. In a rhombus with angles of 124° and 56°, the diagonals split them into 62° + 62° and 28° + 28°, and cross at 90°.
6For a reflex angle, bisect the conjugate angle, then extend the line through the vertex into the reflex part.

Worked example

Problem

Angle XYZ is 136°. Arm YX is 6 cm long and arm YZ is 9 cm long. Bisect the angle using a ruler and compasses, then check it.

⚠ Watch out

Setting the compasses wider than the shorter arm. Then the first arc doesn't cross that arm at all, so there's nowhere to put the compass point next. Any width up to the length of the shorter arm works.

🧠

Memory hook

Same width, rhombus, diagonal. Keep the compasses fixed, the arcs build a rhombus, and its diagonal is your bisector.

✓

Check yourself

Draw any angle and bisect it with only a ruler and compasses. Check it with a protractor. Then explain, in two sentences, why your line has to split the angle exactly in half.

Flashcards

(14)
What does 'bisect' mean?
To cut something into two equal parts. An angle bisector splits an angle into two equal angles.
What is a construction?
A way of making shapes and angles with a ruler and compasses only, without measuring tools.
Is measuring an angle and halving it a construction?
No. It uses a protractor to measure. In a construction, the protractor is only used to check afterwards.
What is a rhombus?
A parallelogram with all four sides the same length. You can make one from two congruent isosceles triangles.
What does a diagonal of a rhombus do to the angles it passes through?
It bisects them: each angle it passes through is cut exactly in half.
A rhombus has angles of 124° and 56°. How do its diagonals split them, and at what angle do the diagonals cross?
124° into 62° + 62°, and 56° into 28° + 28°. The diagonals cross at 90°.
After the first arc, why are the two points on the arms the same distance from the vertex?
Both are on one arc centred on the vertex, and a circle has a constant radius.
What's the widest you can set the compasses?
The length of the shorter arm. Any width shorter than that works too.
Why must every arc use the same compass width?
So all four sides are equal and the shape is a rhombus, not some other quadrilateral.
Why are arcs enough, instead of full circles?
The only points that matter are where an arc crosses an arm or another arc.
Which line is the angle bisector?
The straight line from where the two arcs cross, through the vertex. The last two sides of the rhombus never need drawing.
What happens to construction lines at the end?
They stay. Don't rub them out.
How do you check a bisector?
Measure the two new angles with a protractor. Each should be exactly half of the original angle.
How do you bisect a reflex angle?
Bisect the conjugate angle (the smaller angle between the same two arms), then extend the line through the vertex into the reflex part.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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