Angles in Triangles and Quadrilaterals: KS3 Maths Guide

The interior angles of any triangle always add up to 180°, and the interior angles of any quadrilateral always add up to 360°. These two facts let you find any missing angle in a triangle or four-sided shape, as long as the other angles are known.

Angles in a triangle

The triangle angle-sum rule

The three interior angles of any triangle add up to 180°.

This holds for all triangles — scalene, isosceles, equilateral, right-angled, or obtuse.

angle A + angle B + angle C = 180°

Types of triangle and their angle properties

Triangle type | Sides | Angles
Scalene | All different | All different
Isosceles | Two equal | Two base angles equal
Equilateral | All equal | All 60°
Right-angled | Any lengths | One angle exactly 90°

The isosceles triangle property is especially useful: if you know the triangle is isosceles and one of the base angles, you know the other base angle immediately.

Worked example 1: find a missing angle

A triangle has angles 47° and 68°. Find the third angle.

47° + 68° + c = 180°
115° + c = 180°
c = 65°

Answer: 65° (acute — reasonable)

Worked example 2: isosceles triangle

An isosceles triangle has one angle of 40° at the apex (the angle between the two equal sides). Find the two base angles.

The two base angles are equal. Let each base angle = b.

40° + b + b = 180°
2b = 140°
b = 70°

Answer: each base angle = 70°

Worked example 3: algebra in triangles

The angles of a triangle are x°, (2x + 10)°, and (x − 5)°. Find x and all three angles.

x + (2x + 10) + (x − 5) = 180
4x + 5 = 180
4x = 175
x = 43.75°

Angles: 43.75°, 2(43.75) + 10 = 97.5°, 43.75 − 5 = 38.75°

Check: 43.75 + 97.5 + 38.75 = 180° ✓

Answer: x = 43.75, angles are 43.75°, 97.5°, and 38.75°

The exterior angle of a triangle

An exterior angle is formed by extending one side of a triangle beyond the vertex.

Rule: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles (the two opposite interior angles).

This follows directly from the 180° angle sum: interior angle + exterior angle = 180° (straight line), and all three interior angles = 180°, so the exterior angle must equal the sum of the other two.

Worked example 4

A triangle has interior angles 55° and 72°. A side is extended to form an exterior angle e. Find e.

e = 55° + 72° = 127°

Check: e + third interior angle = 180°. Third interior angle = 180 − 55 − 72 = 53°. 127 + 53 = 180° ✓

Answer: e = 127°

Angles in a quadrilateral

The quadrilateral angle-sum rule

The four interior angles of any quadrilateral add up to 360°.

You can see why: any quadrilateral can be split into two triangles by drawing one diagonal. Each triangle has angles summing to 180°, giving 2 × 180° = 360° in total.

angle A + angle B + angle C + angle D = 360°

Special quadrilaterals and their angle properties

Shape | Angle properties
Square | All angles 90°
Rectangle | All angles 90°
Rhombus | Opposite angles equal; adjacent angles supplementary (add to 180°)
Parallelogram | Opposite angles equal; adjacent angles supplementary
Trapezium | Co-interior angles (same side) supplementary
Kite | One pair of opposite angles equal

Worked example 5: find a missing angle in a quadrilateral

A quadrilateral has angles 85°, 110°, 95°, and d. Find d.

85° + 110° + 95° + d = 360°
290° + d = 360°
d = 70°

Answer: 70° (acute — makes sense)

Worked example 6: parallelogram

A parallelogram has one angle of 65°. Find the other three angles.

In a parallelogram, opposite angles are equal and adjacent angles are supplementary:

Opposite angle to 65° = 65°

Both adjacent angles = 180° − 65° = 115°

The four angles are: 65°, 115°, 65°, 115°

Check: 65 + 115 + 65 + 115 = 360° ✓

Worked example 7: algebra in a quadrilateral

The angles of a quadrilateral are p°, 2p°, (p + 30)°, and 90°. Find p and each angle.

p + 2p + (p + 30) + 90 = 360
4p + 120 = 360
4p = 240
p = 60

Angles: 60°, 120°, 90°, 90°

Check: 60 + 120 + 90 + 90 = 360° ✓

Extending the rule: angle sums in any polygon

The angle-sum formula for any polygon with n sides is:

Sum of interior angles = (n − 2) × 180°

Shape | n | Angle sum
Triangle | 3 | (3 − 2) × 180 = 180°
Quadrilateral | 4 | (4 − 2) × 180 = 360°
Pentagon | 5 | (5 − 2) × 180 = 540°
Hexagon | 6 | (6 − 2) × 180 = 720°

Triangles and quadrilaterals are just the first two cases of this general rule.

Common mistakes to avoid

Mistake 1 — Using 360° for a triangle.
The triangle sum is 180°, not 360°. A quadrilateral sums to 360°.

Mistake 2 — Forgetting that isosceles means TWO equal angles, not three.
Equilateral triangles have three equal angles; isosceles triangles have exactly two equal base angles.

Mistake 3 — Assuming all quadrilaterals have right angles.
Only squares and rectangles have four right angles. A general quadrilateral can have any angles, as long as they sum to 360°.

Mistake 4 — Confusing interior and exterior angles.
The exterior angle of a triangle is formed outside the triangle by extending a side. It is not the same as one of the interior angles.

How polygon angle sums fit the KS3 national curriculum

The Department for Education's KS3 mathematics programme of study requires pupils to "derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures using appropriate language and technologies." The angle-sum rules are among the most frequently tested properties at KS3, and form part of the statutory curriculum content for Key Stage 3 and 4 published by the Department for Education.

Frequently asked questions

Why do the angles in a triangle always add up to 180°?

You can demonstrate this physically: tear the three corners off a triangle and place them side by side along a straight edge. They form a straight line — 180°. The formal proof uses the parallel lines through a vertex: the three angles of the triangle correspond to three angles on a straight line (alternate and corresponding angle rules from parallel line theory), which together make 180°.

Can a triangle have two obtuse angles?

No. If two angles were each greater than 90°, their sum alone would exceed 180°, leaving no room for a positive third angle. Every triangle must have at least two acute angles. The most it can have is one right angle or one obtuse angle.

What is the difference between interior and exterior angles of a polygon?

An interior angle is the angle inside the shape at a vertex, between two adjacent sides. An exterior angle is formed by extending one side past the vertex and measuring the angle between the extension and the next side. For a convex polygon, each exterior angle = 180° − interior angle. The sum of all exterior angles of any convex polygon is always 360°.

How do I find a missing angle when the quadrilateral is irregular?

Apply the same rule: all four interior angles add up to 360°. Add the three known angles and subtract from 360°. The shape being irregular does not change the angle sum — it holds for any quadrilateral, regular or not.

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