Practice Maths

L21 — Calculating Angle Sums

Key Terms

angle sum (triangle)
The three interior angles of any triangle always add to 180°.
angle sum (quadrilateral)
The four interior angles of any quadrilateral always add to 360°.
interior angle
An angle inside a polygon, measured between two adjacent sides.
exterior angle
The angle formed outside a triangle when one side is extended. It equals the sum of the two non-adjacent interior angles.
isosceles angle property
In an isosceles triangle, the two base angles (opposite the equal sides) are always equal.

The Two Golden Rules

  • Triangles: Interior angle sum = 180°
  • Quadrilaterals: Interior angle sum = 360° (= two triangles × 180°)
Finding a Missing Angle: Missing angle = Total sum − (sum of all known angles). Always write the equation first: x = 180 − 70 − 50, not just the answer.

Worked Example — Triangle

A triangle has angles 65° and 48°. Find the third angle.

Step 1 — Angle sum of a triangle = 180°.

Step 2 — Known angles: 65° + 48° = 113°.

Step 3 — Missing angle: 180° − 113° = 67°.

Worked Example — Quadrilateral

A quadrilateral has angles 110°, 80°, and 95°. Find the fourth angle.

Step 1 — Angle sum of a quadrilateral = 360°.

Step 2 — Known angles: 110° + 80° + 95° = 285°.

Step 3 — Missing angle: 360° − 285° = 75°.

Isosceles Triangle Property

In an isosceles triangle, the two base angles are always equal. If the top angle is known, the two base angles are each (180° − top angle) ÷ 2.

Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles. E.g., exterior angle = 40° + 65° = 105° (if the interior angles at the other two vertices are 40° and 65°).

The Triangle Angle Sum Rule

The three interior angles of any triangle always add up to 180°. This is one of the most fundamental rules in geometry. It doesn’t matter if the triangle is tiny or huge, right-angled or equilateral — the interior angles always sum to exactly 180°.

To find a missing angle:

  • Write the equation: x + known1 + known2 = 180
  • Substitute known values: x + 70 + 50 = 180
  • Solve: x = 180 − 70 − 50 = 60°
Always write the angle sum equation before substituting values. This keeps your working clear and helps you catch errors.

The Quadrilateral Angle Sum Rule

The four interior angles of any quadrilateral always add up to 360°. This makes sense because any quadrilateral can be split into two triangles by drawing a diagonal: 2 × 180° = 360°.

This rule applies to all quadrilaterals — rectangles, parallelograms, trapeziums, kites — no exceptions.

Isosceles Triangles

An isosceles triangle has two equal sides and two equal base angles. Use this to your advantage: if you know the top angle, the two base angles are each (180 − top angle) ÷ 2. If you know one base angle, the top angle is 180 − 2 × base.

Exterior Angles

When you extend one side of a triangle beyond a vertex, you create an exterior angle outside the triangle. The Exterior Angle Theorem states that this exterior angle equals the sum of the two non-adjacent interior angles (the angles at the other two vertices).

  • Interior angles: 40°, 65°, 75°. Exterior angle at the 75° vertex = 40° + 65° = 105°.
  • Check: interior + exterior at same vertex = 180° (straight line): 75° + 105° = 180° ✓

Problem-Solving Strategy

When working with angle diagrams:

  • Identify the polygon type (triangle = 180°, quadrilateral = 360°).
  • Look for special properties: right angle, equal sides (isosceles), equilateral (all 60°).
  • Write the equation, substitute knowns, solve for the unknown.
If a diagram has a small square in a corner, that angle is exactly 90°. Don’t re-label it as “x” — use 90 in your equation.

  1. Triangle Warm-up

    Find the missing angle x in each triangle.

    70° 50° x (a)
    110° 25° x (b)
    42° 88° x (c)
    15.5° 100.5° x (d)
    1. Find x in triangle (a).
    2. Find x in triangle (b). What type of triangle is it by angles?
    3. Find x in triangle (c).
    4. Find x in triangle (d). Give your answer as a decimal.
    5. A triangle has angles in the ratio 1 : 2 : 3. Find all three angles, then classify the triangle by its angles.

  2. The Right-Angled Property

    Each triangle contains a right angle (90°). Find the missing angle x.

    30° x (a)
    x x (b)
    12° x (c)
    1. Find x in triangle (a).
    2. In triangle (b), both acute angles are equal (both labelled x). Find x. What is the name of this triangle, classified by its sides?
    3. Find x in triangle (c). What angle type is x?
    4. Can a triangle have two right angles? Use the angle sum rule to explain why or why not.
    5. A right-angled triangle has one acute angle of 37°. List all three angles of the triangle in order from smallest to largest.

  3. Isosceles Triangle Logic

    In an isosceles triangle, the two base angles are always equal. The top angle is the angle between the two equal sides.

    top base base
    1. The top angle is 80°. Find the size of each base angle.
    2. One base angle is 55°. Find the top angle.
    3. The top angle is 120°. Find the two base angles. What angle type are they?
    4. The base angles are each 60°. Find the top angle. What special triangle is this?
    5. A triangle has angles 3x°, 3x°, and 2x°. Use the angle sum rule to find x, then state all three angles. What type of triangle is it by sides?

  4. The 360° Rule — Quadrilaterals

    Find the missing angle in each quadrilateral.

    x (a)
    120° 60° 120° x (b)
    105° 75° 110° x (c)
    150° 30° 150° x (d)
    1. Find x in quadrilateral (a). What shape is it? What could you predict about x without calculating?
    2. Find x in quadrilateral (b).
    3. Find x in quadrilateral (c).
    4. Find x in quadrilateral (d).
    5. A quadrilateral has three angles of 95°, 115°, and 88°. Find the fourth angle.

  5. Parallelograms

    In a parallelogram, opposite angles are equal and adjacent angles add to 180°.

    1. One angle of a parallelogram is 70°. What is the angle directly opposite it?
    2. One angle is 70°. What is the adjacent angle (the one next to it)?
    3. The sum of two opposite angles in a parallelogram is 200°. Find the size of one of those two angles. Then find each of the other two angles.
    4. If all four angles of a parallelogram are equal, what is each angle? What are two possible names for this shape?

  6. Special Quadrilaterals

    1. A kite has one pair of equal opposite angles (110° each) and a top angle of 40°. Find the bottom angle.
    2. An isosceles trapezium has bottom angles of 75° each. What are the two top angles? Explain using the angle sum.
    3. A kite has angles 100°, 100°, and 80°. Find the fourth angle.
    4. Can a quadrilateral have three right angles and one acute angle? Justify your answer using the angle sum rule.

  7. Speed Round

    1. What is the combined angle sum of 2 triangles joined at a common side?
    2. What is the interior angle sum of a rectangle? Why?
    3. A square is cut diagonally in half to form a triangle. What are the three angles of the triangle?
    4. A triangle has angles 60° + 60° + 60°. What is its name, classified by sides AND by angles?

  8. Visual Challenge

    x 55° Triangle A
    115° 115° y Shape B (Trapezium)
    1. Find x in Triangle A.
    2. Find y in Shape B. (The trapezium has two equal top angles of 115° each, and two equal bottom angles. Use this symmetry.)
    3. If the 55° angle in Triangle A were changed to 45°, what would x become?
    4. Verify: do all four angles of Shape B add to 360°?

  9. The “What if?” Scenarios

    1. If a triangle has angles in the ratio 1 : 2 : 3, what are the three angles? (Hint: let the angles be k, 2k, 3k.)
    2. If a quadrilateral has four equal angles, what is the size of each? What is the name of any such shape?
    3. Can an equilateral triangle have a 90° angle? Justify your answer.
    4. If you know only ONE angle in a square, do you know all four angles? Explain.

  10. Mastery Check

    1. True or false: “Every quadrilateral can be split into exactly two triangles by drawing one diagonal.”
    2. An exterior angle of a triangle is 130°. The two non-adjacent interior angles are equal. Find each of those interior angles.
    3. A triangle has an obtuse angle of 105° and an acute angle of 38°. Find the third angle. Can this triangle be isosceles? Justify.
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