lesson
Construct equal angles
1 · Learn
An angle bisector is a ray that splits an angle into two equal angles. We construct it using two pairs of equal lengths. Start with a drawn angle AOB of less than 180°, with O as its vertex.
With compass centre O, draw one arc crossing the two arms at C and D. This makes OC = OD. Next use the SAME new compass opening at C and D to draw two arcs that intersect inside the angle at P. Choose an opening large enough for these arcs to meet, and choose P away from O. Draw ray OP with a ruler. Leave all construction arcs visible.
Why does it work? Triangles OCP and ODP share OP, have OC = OD from the first arc, and CP = DP from the second pair. They are congruent by SSS, so angle COP equals angle POD. OP is the bisector. Copying equal lengths builds the result; measuring half the angle with a protractor is a check, not this construction.
2 · Worked example
For a 90° angle, the construction makes two 45° angles. The diagram shows C and D each 4 cm from O and CP = DP = 4 cm. Equal lengths establish SSS congruence; we do not decide that OP is correct just because it looks central.
3 · Your turn
Draw a 70° angle with a protractor, then bisect it with ruler and compass using the method above. What angle should each half measure? Name the three matching lengths in the proof.
Check your answer
Each half is 35°. OC = OD, CP = DP, and OP is shared. These establish SSS congruence and hence equal corresponding angles.
4 · Apply your learning
Repeat with an obtuse 120° angle. Keep the two compass openings within each pair equal. Check for two 60° halves and explain why measuring equal distances along the two arms alone is not the completed bisector.
Adult guidance and safety
Demonstrate safe compass handling. Use paper, pencil, ruler and compass; keep construction arcs visible. A protractor can check the result but does not replace a compass construction. A drawing pad can record a sketch, but its freehand lines do not establish compass accuracy.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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