Question
Using only a compass and straightedge (no protractor), construct angles of 60°, 120°, 90°, 30°, and 45°.
Solution — Step by Step
Draw a ray OA. With O as centre and any radius, draw an arc cutting OA at P. With P as centre and the same radius, draw an arc cutting the first arc at Q. Join OQ. Angle POQ = 60°.
Why 60°? Because OP = PQ = OQ (all equal to the compass radius), so triangle OPQ is equilateral. Each angle of an equilateral triangle is 60°.
graph TD
A["60° constructed"] --> B["Bisect 60° → 30°"]
A --> C["Double 60° → 120°"]
C --> D["Bisect angle between 120° and 180° → 150°"]
A --> E["Bisect between 60° and 120° → 90°"]
E --> F["Bisect 90° → 45°"]
B --> G["Bisect 30° → 15°"]120°: After constructing the 60° arc point Q, keep the same radius and draw another arc from Q. This gives 120° from OA.
90°: Construct 60° and 120° arcs. Bisect the angle between them (bisect the 60° gap from 60° to 120°). The bisector is at 90°.
30°: Bisect the 60° angle using the standard angle bisector construction.
45°: Bisect the 90° angle.
To bisect any angle: from the vertex, draw an arc cutting both rays. From these two points, draw arcs of equal radius that intersect. Join the vertex to this intersection point — that is the bisector. This works because the two new arcs create an isosceles triangle, and the line from vertex to the far point is both median and angle bisector.
Why This Works
All compass-straightedge constructions rely on two basic operations: drawing circles (compass) and drawing straight lines (straightedge). The 60° angle comes from the equilateral triangle property. Every other standard angle is derived by doubling or bisecting.
The angles you CAN construct this way: any multiple of 15° (0°, 15°, 30°, 45°, 60°, 75°, 90°, 105°, 120°, 135°, 150°, 165°, 180°). You CANNOT construct 1°, 40°, or 20° with compass and straightedge alone — this is a proven mathematical impossibility.
Alternative Method
For CBSE exams, practice these constructions until they are automatic. The marking scheme gives full marks only if construction arcs are visible. Do NOT erase the arcs — they are your proof of correct construction. Keep them light but visible.
A quick reference for the construction sequence: start with 60° (the mother angle), then derive everything else. The full construction tree is: 60° → 30° (bisect) → 15° (bisect again); 60° → 120° (repeat arc) → 90° (bisect 60°-120° gap) → 45° (bisect 90°).
Common Mistake
Changing the compass radius mid-construction. When constructing 60°, the three arcs (from O and from P) must all use the SAME radius. If you accidentally change the compass width, the triangle is no longer equilateral and the angle will not be 60°. Hold the compass firmly and do not adjust it between steps. Similarly, when bisecting, the two arcs from the ray-points must have equal radius (though this can differ from the first arc’s radius).