What is angles in a polygon?

$interior angle sum = (n - 2) × 180^$

Two angles on a straight line are x and 2x + 30^. Find x. [2 marks]

Each exterior angle of a regular polygon is 24^. Find the number of sides. [2 marks]

SEAB Original-style practice: A regular polygon has an interior angle of 150^. Find the number of sides.

The exterior angle is 180^ - 150^ = 30^, since interior and exterior angles are supplementary. The exterior angles of any polygon sum to 360^, so the number of sides is 360^30^ = 12. The polygon has 12 sides. Markers reward finding the exterior angle, using the 360^ exterior-angle sum, and the correct number of sides.

SEAB Original-style practice: In a triangle ABC, angle A = 2x, angle B = 3x and angle C = x + 20^. Find the value of x and hence the largest angle.

The angles of a triangle sum to 180^: 2x + 3x + (x + 20^) = 180^. Simplify: 6x + 20^ = 180^, so 6x = 160^ and x = 160^6 = 2623^. The angles are A = 5313^, B = 80^ and C = 4623^, so the largest is B = 80^. Markers reward the angle-sum equation, solving for x, and identifying the largest angle.

SingaporeMathsSyllabus dot point

What angle rules govern lines, triangles and polygons, and how do we use them?

Apply angle properties of parallel lines, triangles and polygons, including interior and exterior angle sums, to find unknown angles

A focused answer to the O-Level E-Maths outcome on angle properties. Angles on a line and at a point, parallel-line angles, the angle sum of a triangle, and interior and exterior angles of polygons.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-06

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The answer
Examples in context
Try this

What this dot point is asking

SEAB wants you to apply the angle rules for straight lines, parallel lines, triangles and polygons to calculate unknown angles, giving reasons. Angle chasing is the foundation of all geometric reasoning in the syllabus, including the circle theorems.

The answer

Angles on a line and at a point

Angles on a straight line add to $180^\circ$ , and angles around a point add to $360^\circ$ . Vertically opposite angles, formed where two lines cross, are equal. These three facts settle many simple angle problems immediately.

Parallel-line angles

When a straight line (transversal) crosses two parallel lines:

corresponding angles (in matching positions) are equal,
alternate angles (the Z-shape) are equal,
co-interior angles (the C-shape, on the same side) add to $180^\circ$ .

Quoting the correct one of these three by name earns the reasoning marks.

Angles in a triangle

The interior angles of a triangle add to $180^\circ$ . An exterior angle of a triangle equals the sum of the two opposite interior angles. Special triangles help too: an isosceles triangle has two equal base angles, and an equilateral triangle has three $60^\circ$ angles.

Angles in a polygon

For a polygon with $n$ sides:

\text{interior angle sum} = (n - 2) \times 180^\circ

The exterior angles of any polygon always add to $360^\circ$ . For a regular polygon, each exterior angle is $\dfrac{360^\circ}{n}$ and each interior angle is $180^\circ$ minus that.

Worked example

Two parallel lines are crossed by a transversal. One angle is $70^\circ$ . A triangle is formed below using the transversal and a second line meeting it. Find the marked angle $y$ at the third vertex, where the other two angles of the triangle are $70^\circ$ (alternate) and $55^\circ$ .

Step 1: Use the alternate-angle fact

The angle inside the triangle alternate to the given $70^\circ$ is also $70^\circ$ , by alternate angles between parallel lines.

Step 2: Note the second given angle

The second interior angle of the triangle is $55^\circ$ .

Step 3: Apply the triangle angle sum

y = 180^\circ - 70^\circ - 55^\circ

Step 4: Evaluate

y = 55^\circ

The marked angle is $55^\circ$ , found using alternate angles and the triangle angle sum.

Examples in context

Example 1. Tiling patterns. Regular polygons that tile a floor must have interior angles dividing exactly into $360^\circ$ at each vertex. This is why squares, equilateral triangles and regular hexagons tile, but regular pentagons leave gaps.

Example 2. Surveying directions. Bearings and turns when navigating use angles on a line and around a point. Knowing that a full turn is $360^\circ$ lets a surveyor add up successive turns to track direction.

Try this

Q1. Find the interior angle sum of a hexagon. [1 mark]

Cue. $(6 - 2) \times 180^\circ = 720^\circ$ .

Q2. Two angles on a straight line are $x$ and $2x + 30^\circ$ . Find $x$ . [2 marks]

Cue. $x + 2x + 30^\circ = 180^\circ$ , so $3x = 150^\circ$ and $x = 50^\circ$ .

Q3. Each exterior angle of a regular polygon is $24^\circ$ . Find the number of sides. [2 marks]

Cue. $\dfrac{360^\circ}{24^\circ} = 15$ sides.

Exam-style practice questions

Practice questions written in the style of SEAB exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Original3 marksA regular polygon has an interior angle of

150^\circ

. Find the number of sides.

Show worked answer →

The exterior angle is $180^\circ - 150^\circ = 30^\circ$ , since interior and exterior angles are supplementary.

The exterior angles of any polygon sum to $360^\circ$ , so the number of sides is $\dfrac{360^\circ}{30^\circ} = 12$ .

The polygon has $12$ sides.

Markers reward finding the exterior angle, using the $360^\circ$ exterior-angle sum, and the correct number of sides.

Original3 marksIn a triangle

ABC

, angle

A = 2x

, angle

B = 3x

and angle

C = x + 20^\circ

. Find the value of

x

and hence the largest angle.

Show worked answer →

The angles of a triangle sum to $180^\circ$ : $2x + 3x + (x + 20^\circ) = 180^\circ$ .

Simplify: $6x + 20^\circ = 180^\circ$ , so $6x = 160^\circ$ and $x = \dfrac{160^\circ}{6} = 26\dfrac{2}{3}^\circ$ .

The angles are $A = 53\dfrac{1}{3}^\circ$ , $B = 80^\circ$ and $C = 46\dfrac{2}{3}^\circ$ , so the largest is $B = 80^\circ$ .

Markers reward the angle-sum equation, solving for $x$ , and identifying the largest angle.