Congruent triangles · Ages 13–14
Angles compasses can draw
Start from 60° and 90°: bisect, add, take away.
How to do it
Method
Start from 60° and 90°: bisect, add, take away
75° = 45° + 30°: half of 90° and half of 60°
Another way
Take it from a straight line
150° = 180° − 30°
In full
With compasses and a straight edge alone you can start from two angles: the 60° of an equilateral triangle and the 90° of a perpendicular. For 60°: with centre O, draw an arc cutting the ray OA at P; with centre P and the same radius, draw an arc crossing the first at Q; join OQ, and △OPQ has three equal sides, so ∠AOQ = 60°. For 90°: draw a perpendicular bisector. Bisecting halves an angle: 60° → 30° → 15° → 7.5°, and 90° → 45° → 22.5°. Adding or taking away angles already drawn makes more: 75° = 45° + 30°, 105° = 60° + 45°, 135° = 90° + 45°, 150° = 180° − 30°, 120° = 60° + 60°. One way: break the angle into a sum or difference of these. Another: see whether it is built up from 15° (a multiple of 15°), or half of one of these. Some can't be made by bisecting, adding or taking away, such as 20°, 40° or 50°: cutting 60° into three equal parts can't be done with compasses and a straight edge alone.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
An angle of 70° can be drawn with compasses and a straight edge alone.
True or false?
Show the answer and working
Answer: False
From 60° (an equilateral triangle) and 90° (a perpendicular), bisecting, adding and taking away give multiples of 15° such as 15°, 30°, 45° and 75° = 45° + 30°; angles such as 20°, 40° or 50° can't be made that way.
Example 2
With compasses and a straight edge, a 60° angle is bisected, then the result bisected, and so on: 1 times in all. The last angle is °
Show the answer and working
Answer: 30
Each bisection halves the angle: 60° bisected 1 times gives 30°.
Example 3
A 90° angle is drawn and bisected 1 times; next to it, a 90° angle is drawn and bisected 2 times. Together the two make °
Show the answer and working
Answer: 67.5
90° bisected 1 times is 45°, 90° bisected 2 times is 22.5°, and together 45° + 22.5° = 67.5°.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.How can a 15° angle be drawn with compasses and a straight edge alone?
- A. Draw a 60° angle, then bisect it
- B. Draw an equilateral triangle and use one of its angles
- C. Draw a 30° angle, then bisect it
- D. Draw a right angle (90°), then bisect it
Choose the right answer
Answer:
2.Which of these angles can be drawn with compasses and a straight edge alone?
- A. 20°
- B. 45°
- C. 70°
- D. 40°
Choose the right answer
Answer:
3.Draw a 60° angle ∠AOQ on the ray OA with compasses and a straight edge, the compasses open at 6 cm: put the steps in order.
- A. OP = 6 cm: with centre O and radius 6 cm, draw an arc cutting OA at P
- B. ∠AOQ = 60°: join OQ with the straight edge
- C. PQ = OP: with centre P and the same radius, draw an arc crossing the first at Q
Write the letters in the right order
Answer:
4.A 60° angle is drawn and bisected 1 times; next to it, a 90° angle is drawn and bisected 1 times. Together the two make °
Answer:
5.With compasses and a straight edge, a 90° angle is bisected, then the result bisected, and so on: 1 times in all. The last angle is °
Answer:
All the answers
- 1. C. Draw a 30° angle, then bisect it
- 2. B. 45°
- 3. A → C → B
- 4. 75
- 5. 45
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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