Geometry and proof · Ages 13–14

Angles compasses can draw worksheet

Part of: Congruent triangles

In the print window, choose “Save as PDF” to download it. The answers print on the last page.

Geometry and proof · Ages 13–14

Numsprite
numsprite.com

Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.Draw a 60° angle ∠AOQ on the ray OA with compasses and a straight edge, the compasses open at 8 cm: put the steps in order.
- A. PQ = OP: with centre P and the same radius, draw an arc crossing the first at Q
- B. OP = 8 cm: with centre O and radius 8 cm, draw an arc cutting OA at P
- C. ∠AOQ = 60°: join OQ with the straight edge
Write the letters in the right order
Answer:
2.Draw a 60° angle ∠AOQ on the ray OA with compasses and a straight edge, the compasses open at 5 cm: put the steps in order.
- A. PQ = OP: with centre P and the same radius, draw an arc crossing the first at Q
- B. ∠AOQ = 60°: join OQ with the straight edge
- C. OP = 5 cm: with centre O and radius 5 cm, draw an arc cutting OA at P
Write the letters in the right order
Answer:
3.A 90° angle is drawn and bisected 2 times; next to it, a 90° angle is drawn and bisected 1 times. Together the two make °
Answer:
4.With compasses and a straight edge, a 90° angle is bisected, then the result bisected, and so on: 2 times in all. The last angle is °
Answer:
5.A 60° angle is drawn and bisected 2 times; next to it, a 90° angle is drawn and bisected 1 times. Together the two make °
Answer:
6.To draw 120° with compasses and a straight edge, draw a 90° angle first, then add an angle of ° beside it.
Answer:
7.To draw 105° with compasses and a straight edge, draw a 90° angle first, then add an angle of ° beside it.
Answer:

8.Which of these angles can be drawn with compasses and a straight edge alone?
- A. 20°
- B. 135°
- C. 110°
- D. 10°
Choose the right answer
Answer:
9.An angle of 45° can be drawn with compasses and a straight edge alone.
True or false?
Answer:
10.How can a 45° angle be drawn with compasses and a straight edge alone?
- A. Draw a 60° angle, then bisect it
- B. Draw a right angle (90°), then bisect it
- C. Draw an equilateral triangle and use one of its angles
- D. Draw a 30° angle, then bisect it
Choose the right answer
Answer:
11.An angle of 30° can be drawn with compasses and a straight edge alone.
True or false?
Answer:
12.Which of these angles can't be drawn with compasses and a straight edge alone?
- A. 7.5°
- B. 50°
- C. 135°
- D. 30°
Choose the right answer
Answer:

13.An angle of 67.5° can be drawn from 60° and 90° by bisecting, adding or taking away, with compasses and a straight edge alone.
True or false?
Working:
Answer:
14.At 8 o'clock, what is the angle between a clock's hour and minute hands (the one less than 180°)?
Working:
Answer:
15.To draw 150° with compasses and a straight edge alone, which way works?
- A. Draw 30° (half of 60°) on a straight line; the angle beside it is 180° − 30° = 150°
- B. Draw 90°, then add 50°
- C. Draw 60°, then bisect it twice
- D. Draw 120°, then add 45°
Choose the right answer
Working:
Answer:
16.To draw 157.5°, draw an angle on a straight line first; the angle beside it is 157.5°. The first angle =
Working:
Answer:
17.Bisecting a 90° angle 2 times gives °; bisecting a 60° angle 2 times gives °.
Fill in each blank
Working:
Answer:
18.To draw 135°, draw an angle on a straight line first; the angle beside it is 135°. The first angle =
Working:
Answer:

Show the answers and working

Angles compasses can draw worksheet: Answers and working

1. B → A → C
OQ, PQ and OP all equal the radius, 8 cm, so △OPQ is equilateral and ∠AOQ = 60°.
2. C → A → B
OQ, PQ and OP all equal the radius, 5 cm, so △OPQ is equilateral and ∠AOQ = 60°.
3. 67.5
90° bisected 2 times is 22.5°, 90° bisected 1 times is 45°, and together 22.5° + 45° = 67.5°.
4. 22.5
Each bisection halves the angle: 90° bisected 2 times gives 22.5°.
5. 60
60° bisected 2 times is 15°, 90° bisected 1 times is 45°, and together 15° + 45° = 60°.
6. 30
120° = 90° + 30°, and 30° comes from 60° or 90° by bisecting or adding.
7. 15
105° = 90° + 15°, and 15° comes from 60° or 90° by bisecting or adding.

8. B. 135°
135° is a multiple of 15°, made from 60° and 90° by bisecting, adding or taking away; none of the others is.
9. True
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.
10. B. Draw a right angle (90°), then bisect it
60° comes from an equilateral triangle and 90° from a perpendicular; each bisection halves: 60° → 30° → 15°, and 90° → 45°.
11. True
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.
12. B. 50°
50° can't be made from 60° and 90° by bisecting, adding or taking away; the other three can.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. True
Angles made this way are multiples of 15°, or halves and quarters of them: 67.5° × 2 must be a multiple of 15°.
14. 120°
Neighbouring numbers on a clock face are 360° ÷ 12 = 30° apart; at 8 o'clock the smaller angle between the hands is 120°, a multiple of 15°, which compasses can draw.
15. A. Draw 30° (half of 60°) on a straight line; the angle beside it is 180° − 30° = 150°
150° = 180° − 30°, and 30° is half of 60°; 50° can't be drawn, 60° bisected twice is 15°, and 120° + 45° = 165°.
16. 22.5°
Adjacent angles on a straight line add to 180°: 180° − 157.5° = 22.5°.
17. 22.5, 15
Each bisection halves the angle: 90° bisected 2 times is 22.5°, 60° bisected 2 times is 15°.
18. 45°
Adjacent angles on a straight line add to 180°: 180° − 135° = 45°.

Numsprite · numsprite.com/en/worksheets/congruent-triangles/angles-constructed-with-compasses