Mara is designing a stained-glass panel. Every glass piece she cuts is a regular polygon. (a) One piece is a regular polygon whose interior angle is 150°. Work out the number of sides of this piece. [2] (b) A second piece is a regular polygon with 9 sides. Calculate the size of…

(a) 12 sides; (b) 140°; (c) 18 sides.

Angles and Polygons: O-Level / IGCSE Maths (0580)

Syllabus C4.6, E4.6 · Strand 4 Geometry

Questions: 3
Total marks: 13
Tier mix: 2 Core · 1 Extended

This topic brings together the angle facts you need to find unknown angles in a wide range of geometric figures. You will work with angles meeting at a point (which sum to $360^\circ$ ), angles on a straight line (which sum to $180^\circ$ ), and vertically opposite angles formed where two lines cross. These basic relationships are then extended to the angles inside a triangle, which add to $180^\circ$ , and the angles inside a quadrilateral, which add to $360^\circ$ .

A second key skill is reasoning about parallel lines cut by a transversal. Here you identify and use corresponding angles (equal), alternate angles (equal), and co-interior or allied angles (which sum to $180^\circ$ ) to track an angle from one line to another. The topic also covers polygons: for a polygon with $n$ sides the interior angles sum to $(n-2)\times 180^\circ$ , while the exterior angles always sum to $360^\circ$ . For a regular polygon each exterior angle is $\frac{360^\circ}{n}$ , and each interior angle is its supplement, $180^\circ-\frac{360^\circ}{n}$ . The usual method is to set up an equation from the relevant angle fact, then solve for the unknown, often quoting the reason at each step.

The worked examples below are original, written from the syllabus objective to show these methods in exam-style settings with full worked solutions.

Question 1

Structured Core 4 marks

Two parallel lines $PQ$ and $RS$ are drawn, with $PQ$ above $RS$ .

A straight line (a transversal) crosses the upper line $PQ$ at the point $A$ and crosses the lower line $RS$ at the point $B$ . A third point $C$ lies on the lower line $RS$ , to the right of $B$ , so that $A$ , $B$ and $C$ form triangle $ABC$ .

At $A$ , the angle measured from $PQ$ (towards $Q$ , the left-hand end) to the line $AB$ is angle $QAB = 58^\circ$ .

At $C$ , the angle of the triangle, angle $ACB$ , is $47^\circ$ .

(The two parallel lines are $PQ$ and $RS$ ; $AB$ is the transversal cutting them, forming a "Z" shape with $A$ on $PQ$ and $B$ on $RS$ .)

(a) Write down the size of angle $ABC$ , the angle between the line $RS$ and the line $BA$ . Give a geometrical reason for your answer. [2]

(b) Work out the size of angle $BAC$ . Give a geometrical reason for your answer. [2]

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Question 2

Structured Extended 7 marks

Mara is designing a stained-glass panel. Every glass piece she cuts is a regular polygon.

(a) One piece is a regular polygon whose interior angle is $150^\circ$ . Work out the number of sides of this piece. [2]

(b) A second piece is a regular polygon with $9$ sides. Calculate the size of each interior angle of this piece. [2]

(c) A third piece is a regular polygon in which each interior angle is $8$ times the size of each exterior angle. Find the number of sides of this piece. [3]

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Question 3

Multiple choice Core 2 marks

$ABCDE$ is a regular pentagon. The side $AB$ is produced (extended) beyond $B$ to a point $F$ , so that $A$ , $B$ and $F$ lie on a straight line. Calculate the size of angle $CBF$ .

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