Let’s go through each angle construction step by step, using only a compass and straightedge (ruler without markings), as shown in the image. We’ll explain how to build each one from scratch — starting with 60°, then bisecting or combining angles to get others.
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Step 1: Construct ∠AOB = 60°
- Draw a ray OA.
- Place compass at O, draw an arc that cuts OA at point P.
- Without changing compass width, place compass at P, draw another arc that intersects the first arc at Q.
- Draw ray OB through Q.
→ ∠AOB = 60° (because triangle OPQ is equilateral).
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Step 2: Construct ∠AOC = 30°
- Start with ∠AOB = 60°.
- Bisect it:
- From points where the arc cuts OA and OB, draw two arcs inside the angle that intersect.
- Draw ray OC through that intersection.
→ ∠AOC = 30° (half of 60°).
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Step 3: Construct ∠AOD = 15°
- Start with ∠AOC = 30°.
- Bisect it again:
- Same method: draw arcs from both sides of the 30° angle, find their intersection, draw ray OD.
→ ∠AOD = 15° (half of 30°).
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Step 4: Construct ∠AOE = 120°
- Start with ∠AOB = 60°.
- Extend the arc beyond B, and mark another 60° from B (same compass width) to get point E.
- So from OA to OE is 60° + 60° = 120°.
→ ∠AOE = 120°.
*(Alternatively: construct 60°, then add another 60° adjacent to it.)*
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Step 5: Construct ∠AOF = 180°
- Just extend ray OA backwards through O to form a straight line.
→ ∠AOF = 180° (straight angle).
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Step 6: Construct ∠AOG = 90°
- First construct 60° (∠AOB).
- Then construct 120° (∠AOE) as above.
- Bisect the angle between 60° and 120° → that’s 90°.
OR simpler:
- Draw a semicircle on OA extended to F (180°).
- With same radius, mark arcs from A and F — they meet above O → connect to O for 90°.
→ ∠AOG = 90°.
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Step 7: Construct ∠AOH = 45°
- Start with ∠AOG = 90°.
- Bisect it → you get 45°.
→ ∠AOH = 45°.
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Step 8: Construct ∠AOI = 22½°
- Start with ∠AOH = 45°.
- Bisect it → 22.5°.
→ ∠AOI = 22½°.
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Step 9: Construct ∠AOJ = 75°
- You can do this by adding 60° + 15°.
- Construct 60° (∠AOB).
- Construct 15° (∠AOD) next to it (on the other side of OB).
- Total from OA to OJ = 60° + 15° = 75°.
OR:
- Construct 90°, then subtract 15° (bisect 30° twice).
→ ∠AOJ = 75°.
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Step 10: Construct ∠AOK = 37½°
- This is half of 75°.
- So first make 75° (∠AOJ), then bisect it.
→ ∠AOK = 37½°.
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Step 11: Construct ∠AOL = 150°
- 150° = 180° – 30°.
- Draw straight line AOF (180°).
- Construct 30° downward from OF (or upward from OA, but on the other side).
- Or: 90° + 60° = 150°.
→ ∠AOL = 150°.
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Step 12: Construct ∠AOM = 135°
- 135° = 90° + 45°.
- Construct 90° (∠AOG), then add 45° next to it.
OR:
- 180° – 45° = 135°.
→ ∠AOM = 135°.
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All these constructions are built using just two basic moves:
1. Making a 60° angle (using equilateral triangle property).
2. Bisecting any angle (by drawing intersecting arcs from both rays).
You can combine and halve angles to get many more!
Final Answer:
The constructions for all listed angles (60°, 30°, 15°, 120°, 180°, 90°, 45°, 22½°, 75°, 37½°, 150°, 135°) are achieved by repeatedly constructing 60° angles and bisecting them, or adding/subtracting known angles using compass and straightedge techniques.
Parent Tip: Review the logic above to help your child master the concept of construction of angles in geometry worksheet for practice.
