More angles with a compass
Draw BC = 6 cm and construct ∠ABC = 15° at B.
Draw BC = 5 cm and construct ∠ABC = 75° at B.
Draw BC = 4.5 cm and construct ∠ABC = 105° at B.
Draw BC = 5 cm and construct ∠ABC = 135° at B.
Draw BC = 5 cm and construct ∠ABC = 150° at B.
1. Draw BC = 6 cm.
2. With B as the center and any radius (here 2 cm), draw an arc that cuts BC at P.
3. With P as the center and the same radius, cut the arc at Q.
4. With P and Q as the centers and the same radius, draw arcs that cut at S.
5. Join BS. It cuts the first arc at M, and ∠MBC = 30°.
6. With P and M as the centers and the same radius, draw arcs that cut at U.
7. Join BU and produce it to A: BU bisects ∠MBC = 30°, so ∠ABC = 15°.
1. Draw BC = 5 cm.
2. With B as the center and any radius (here 2 cm), draw an arc that cuts BC at P.
3. With P as the center and the same radius, cut the arc at Q.
4. With Q as the center and the same radius, cut the arc again at R.
5. With Q and R as the centers and the same radius, draw arcs that cut at S.
6. Join BS. It cuts the first arc at T, and ∠TBC = 90°.
7. With Q and T as the centers and the same radius, draw arcs that cut at U.
8. Join BU and produce it to A: BU bisects ∠TBQ, so ∠ABC = 60° + 15° = 75°.
1. Draw BC = 4.5 cm.
2. With B as the center and any radius (here 2 cm), draw an arc that cuts BC at P.
3. With P as the center and the same radius, cut the arc at Q.
4. With Q as the center and the same radius, cut the arc again at R.
5. With Q and R as the centers and the same radius, draw arcs that cut at S.
6. Join BS. It cuts the first arc at T, and ∠TBC = 90°.
7. With T and R as the centers and the same radius, draw arcs that cut at U.
8. Join BU and produce it to A: BU bisects ∠RBT, so ∠ABC = 90° + 15° = 105°.
1. Draw BC = 5 cm and produce it a little beyond B.
2. With B as the center and any radius (here 2 cm), draw an arc that cuts BC at P and CB produced at W.
3. With P as the center and the same radius, cut the arc at Q.
4. With Q as the center and the same radius, cut the arc again at R.
5. With Q and R as the centers and the same radius, draw arcs that cut at S.
6. Join BS. It cuts the first arc at T, and ∠TBC = 90°.
7. With T and W as the centers and the same radius, draw arcs that cut at U.
8. Join BU and produce it to A: BU bisects ∠WBT, so ∠ABC = 90° + 45° = 135°.
1. Draw BC = 5 cm and produce it a little beyond B.
2. With B as the center and any radius (here 2 cm), draw an arc that cuts BC at P and CB produced at W.
3. With P as the center and the same radius, cut the arc at Q.
4. With Q as the center and the same radius, cut the arc again at R.
5. With R and W as the centers and the same radius, draw arcs that cut at U.
6. Join BU and produce it to A: BU bisects ∠WBR = 60°, so ∠ABC = 120° + 30° = 150°.
Why it works
Every arc has the same radius, so B, P and Q are the corners of an equilateral triangle, and ∠QBP = 60°. Each further cut adds another 60°. Arcs of the same radius from two points on the arc meet on the line that bisects the angle between them: halving gives 30° from 60°, 90° between 60° and 120°, and so on.