Constructing Angles of 15° and 75°

1. Building new angles from old ones

You can already construct 60°, 30°, 90° and 45°. A clever builder does not learn a new method for every angle. She uses two ideas over and over: bisect an angle to halve it, and add or subtract angles that are next to each other. With these two ideas you can make 15° and 75° as well.

Notice how neatly the angles link together: 60° → 30° → 15° → 7½°. Each is exactly half of the one before. So 15° needs three constructions: 60°, then bisect to 30°, then bisect to 15°. These are the three quick constructions from the last lessons.

2. Constructing 15°

The curriculum says: Construct an angle of 15° by bisecting an angle of 30°, which in turn is got by bisecting 60°. Follow the steps carefully with a sharp pencil. Because you bisect twice, small errors can grow, so draw your arcs lightly and accurately.

1. 1Construct angle AOB = 60° (an arc from O cuts the line at A; the same radius from A cuts the arc at B).
2. 2Bisect angle AOB: arcs from A and B (same opening) cross at P. Join OP. Angle AOP = 30°.
3. 3Bisect angle AOP: draw an arc from the point where OP meets the first arc and another from A. They cross at G.
4. 4Join O to G. Angle AOG = 30° ÷ 2 = 15°.

▶ Watch the angle shrink: 60° to 30° to 15°. Each bisecting uses two equal arcs.

To verify, place the protractor on O. The line OG should be at 15°. If the reading is 14° or 16°, that is fine. If it is 12° or 18°, check your arcs.

3. Constructing 75°

Seventy-five degrees is a very useful angle: it is 60° + 15°, or 90° − 15°. The curriculum method uses the right angle: construct 90° at O, construct 60° beside it, and then bisect the gap between 60° and 90°.

1. 1At O, construct 60° (point B on the arc) and 90° (point D on the arc). The angle between OB and OD is 30°.
2. 2Bisect angle BOD: with the compass on B and then on D (same opening), draw arcs that cross at F.
3. 3Join O to F. Because OF lies half way between 60° and 90°, angle AOF = 60° + 15° = 75°.

▶ Notice that 75° = 60° + 15°: the bisector OF sits half way between the 60° line and the 90° line.

Constructing and bisecting 75°. Curriculum example 4 asks you to construct 75° and then bisect it. Bisecting 75° gives 75° ÷ 2 = 37.5° (that is 37½°). Use the standard bisector steps from the vertex.

4. Checking, combining and using

Because 75° and 15° add up to 90°, they are complementary angles. This gives a quick check: if you have constructed 75° and then 15° on the other side of the same arm, the two together must form a right angle.

Another useful fact: a clock face is divided into 12 hours, so each hour mark is 360° ÷ 12 = 30° from the next. Bisecting that gives 15°, the angle between the hour hand and the next half-hour position.

Worked examples

Try each one yourself first, then open the solution and compare every step.