Exploring Some Geometric Themes

Overview

This chapter explores geometric constructions and themes such as symmetry, angles formed by lines, and constructing quadrilaterals with given measurements using ruler and compass.

Key concepts

• A unique quadrilateral needs five independent measurements.
• Constructions use only a ruler and a pair of compasses.
• Angles on a line add to 180°; angles around a point add to 360°.
• Symmetry includes line (reflection) and rotational symmetry.

Important formulae

• Linear pair: angle 1 + angle 2 = 180°.
• Vertically opposite angles are equal.
• Order of rotational symmetry = 360° ÷ smallest angle of rotation.

Solved example

1. Two angles form a linear pair; one is 110°. Find the other.
2. A linear pair sums to 180°.
3. Other angle = 180° − 110° = 70°.

Important questions

1. Construct a quadrilateral PQRS with PQ = 4 cm, QR = 5 cm, RS = 6 cm, PS = 5.5 cm and PR = 7 cm.
2. Two lines intersect; one angle is 65°. Find the other three angles.
3. State the order of rotational symmetry of a square.
4. Draw the lines of symmetry of a regular hexagon and count them.

Quick revision

Five measurements fix a quadrilateral for construction. Use ruler and compass carefully. Remember angle relations at lines and points, and identify both line and rotational symmetry in figures.