Revision notes
Construct 60°, 90° and their bisectors (30°, 45°, 15°…) with only a ruler and a pair of compasses.
Long before protractors, builders marked perfect right angles with a rope, a peg and a straight edge. In your maths set, the pair of compasses does the same job — and it is more accurate than a protractor.
To construct an angle, you may use only a ruler (for straight lines) and a pair of compasses (for arcs). No protractor!
Keep your pencil sharp and do not rub out your arcs — they show your working.
🔑 Key idea: Constructions use only a ruler and a pair of compasses.
Draw a line and mark a point A. With the compass point on A, draw an arc that cuts the line at B.
Keep the same radius. Put the compass point on B and draw an arc that cuts the first arc at C. Join A to C.
AB, BC and AC all equal the radius, so triangle ABC is equilateral, and every angle of an equilateral triangle is 180° ÷ 3 = 60°.
🔑 Key idea: Same radius twice → equilateral triangle → 60°.
To bisect an angle is to cut it into two equal halves. With the compass on the vertex, draw an arc cutting both arms at P and Q. From P and from Q, draw arcs (same radius) that cross at R. Join the vertex to R.
Bisecting 60° gives 30°; bisecting 30° gives 15°.
A straight line is 180°. Bisecting it at a point P gives 90°. Bisecting 90° gives 45°, and bisecting the angle between 60° and 90° gives (60° + 90°) ÷ 2 = 75°.
🔑 Key idea: Start from 60° and 90°, then bisect or combine to get 30°, 45°, 15°, 75°, 120°, 135°…
Which of these gives an angle of 15°, using only a ruler and a pair of compasses?
Step 1: Halfway between two angles
(0° + 30°) ÷ 2 = 15°
Why? Bisecting 30° halves it.
Answer: Bisect an angle of 30°
Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.