Constructions (Using Ruler and Compass only)

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Steps: 1. Draw a line segment EF of any suitable size. 2. With B as centre, draw an arc of any suitable radius which cuts AB at P and BC at Q. 3. With E as centre and the same radius, draw an arc whic…

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The same arc radius is used from the original angle and the new line, so the distances between the cut points are copied exactly. Then the second arc uses PQ as the fixed distance, so the new angle ha…

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Formula: ∠ABP = ∠PBC = 1/2 ∠ABC. Worked use: 1. A ray BP bisects ∠ABC. 2. So the two smaller angles are equal. 3. Each part is half of the full angle. Example form: if ∠ABC is split by BP, then ∠ABP =…

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Steps: 1. With B as centre, draw an arc of any suitable radius which cuts AB at D and BC at E. 2. Taking D and E as centres, draw arcs of equal radii and let them cut each other at O. 3. Join BO and p…

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If the radius is too small, the arcs from D and E may not meet. The radius must be more than half the distance between D and E so that the two arcs intersect and the bisector can be drawn correctly.

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Steps: 1. Draw a line segment BC of any suitable length. 2. With B as centre, draw an arc of any suitable radius which cuts BC at D. 3. With D as centre and the same radius, draw another arc which cut…

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A 60° angle is made directly using two equal-radius arcs. It is not formed by bisecting another angle. The bisection steps are used for 30° and 45°, not for 60°.

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Steps: 1. Draw a 60° angle. 2. Draw BP, the bisector of ∠ABC. 3. Then the smaller angle is half of 60°. Calculation: ∠PBC = 1/2 ∠ABC = 1/2 × 60° = 30°.