Geometry Worked Examples for IGCSE Maths

Question

A, B, C, D are points on a circle with centre O. Angle AOB = 130 degrees. Find angle ACB.

Solution

1. 1

   Identify the theorem

   Angle AOB is at the centre, angle ACB is at the circumference, both subtended by arc AB.

   The angle at the centre is twice the angle at the circumference when subtended by the same arc.

2. 2

   Apply the theorem

   Angle ACB = 130/2 = 65 degrees

   The circumference angle is half the central angle.

Question

Triangles ABC and DEF are similar. AB = 6 cm, BC = 8 cm, AC = 10 cm, and DE = 9 cm. Find EF.

Solution

1. 1

   Find the scale factor

   AB corresponds to DE. Scale factor = DE/AB = 9/6 = 1.5

   Divide corresponding sides to find the scale factor.

2. 2

   Apply the scale factor

   EF = BC x 1.5 = 8 x 1.5 = 12 cm

   Multiply the corresponding side by the scale factor.

Question

Two parallel lines are cut by a transversal. Angle a = 72 degrees. Find angles b (alternate), c (co-interior), and d (corresponding).

Solution

1. 1

   Find alternate angle b

   b = 72 degrees (alternate angles are equal)

   Alternate angles form a Z-shape between parallel lines.

2. 2

   Find co-interior angle c

   c = 180 - 72 = 108 degrees (co-interior angles sum to 180)

   Co-interior (or allied) angles form a C or U shape and add up to 180.

3. 3

   Find corresponding angle d

   d = 72 degrees (corresponding angles are equal)

   Corresponding angles are in the same position at each intersection (F-shape).

Question

A regular polygon has an exterior angle of 30 degrees. How many sides does it have?

Solution

1. 1

   Use the exterior angle formula

   Number of sides = 360 / exterior angle = 360 / 30 = 12

   The exterior angles of any polygon sum to 360 degrees. For a regular polygon, all exterior angles are equal.