Topic 6 of 11 · Form 1
Geometry: angles, polygons, circles and constructions
Angles on a straight line (180°), at a point (360°), vertically opposite, corresponding and alternate angles (parallel lines), angles in triangles and quadrilaterals.
Notes for this topic 1 file, in order
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Angle facts
Angles on a straight line (180°), at a point (360°), vertically opposite, corresponding and alternate angles (parallel lines), angles in triangles and quadrilaterals.
Polygons
Interior angle sum = (n − 2) × 180°; each exterior angle of a regular polygon = 360° ÷ n.
Worked example: Each interior angle of a regular polygon is 156°. How many sides has it?
Exterior angle = 24° → n = 360 ÷ 24 = 15
Circle theorems
Angle in a semicircle = 90°; angles in the same segment are equal; angle at the centre = twice the angle at the circumference; opposite angles of a cyclic quadrilateral sum to 180°; tangent ⟂ radius; tangents from a point are equal; alternate segment theorem.
Symmetry, congruence and similarity
Line and rotational symmetry; congruent triangles (SSS, SAS, ASA, RHS); similar shapes – lengths scale by k, areas by k², volumes by k³.
Worked example: Two similar cylinders have heights 4 cm and 6 cm. The smaller has volume 64 cm³. Find the larger volume.
k = 1.5 → volume = 64 × 1.5³ = 216 cm³
Constructions and loci
Bisectors, perpendiculars, angles, and loci of points a fixed distance from a point or line, or equidistant from two points or lines.
Check yourself
- Find the sum of the interior angles of a hexagon.
- The angle at the centre of a circle is 110°. Find the angle at the circumference on the same arc.
- Two similar shapes have area ratio 9 : 25. Find the length ratio.
Show answers
- 720°
- 55°
- 3 : 5
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