When angles in a diagram are expressed as algebraic expressions, you use a known angle fact — such as angles in a triangle summing to 180° — to set up an equation. Solving that equation gives the value of the unknown, and you can then find each angle.
What angle facts do you need to know?
Before writing any equation, you must recall the relevant angle rule. The six most useful rules at KS3 are:
| Angle fact | Sum |
|---|---|
| Angles on a straight line | 180° |
| Angles around a point | 360° |
| Angles in a triangle | 180° |
| Angles in a quadrilateral | 360° |
| Vertically opposite angles | Equal |
| Corresponding / alternate angles (parallel lines) | Equal |
The question will contain enough information for exactly one of these rules — identify which one, write the equation, then solve.
How do you find angles on a straight line?
Worked example 1: Three angles on a straight line are (2x + 10)°, (x − 5)° and 45°. Find x and each angle.
Step 1 — write the equation: (2x + 10) + (x − 5) + 45 = 180
Step 2 — collect like terms: 3x + 50 = 180
Step 3 — solve: 3x = 130 → x = 130/3 ≈ 43.3°
That gives a non-integer answer, so let us adjust the example for clarity.
Worked example 1 (adjusted): Angles (3x + 5)°, (2x)° and (x − 5)° lie on a straight line.
(3x + 5) + 2x + (x − 5) = 180 6x = 180 x = 30
Angles: 3(30) + 5 = 95°, 2(30) = 60°, 30 − 5 = 25°
Check: 95 + 60 + 25 = 180 ✓
How do you find angles in a triangle?
Worked example 2: A triangle has angles (4x)°, (2x + 18)° and (x − 3)°. Find x and the three angles.
Step 1: Angle sum of a triangle = 180°. 4x + (2x + 18) + (x − 3) = 180
Step 2: Collect: 7x + 15 = 180
Step 3: Solve: 7x = 165 → x = 165/7 ≈ 23.6°
Again non-integer — let me use a cleaner version.
Worked example 2 (clean): Angles in a triangle are (3x + 10)°, (2x − 5)° and 45°.
(3x + 10) + (2x − 5) + 45 = 180 5x + 50 = 180 5x = 130 x = 26
Angles: 3(26) + 10 = 88°, 2(26) − 5 = 47°, 45°
Check: 88 + 47 + 45 = 180 ✓
How do you find angles around a point?
Worked example 3: Four angles around a point are (5x)°, (3x + 20)°, (2x − 10)° and 90°.
Equation: 5x + (3x + 20) + (2x − 10) + 90 = 360 10x + 100 = 360 10x = 260 x = 26
Angles: 130°, 98°, 42°, 90°
Check: 130 + 98 + 42 + 90 = 360 ✓
How do you use vertically opposite or parallel-line angle rules?
Vertically opposite example: Two vertically opposite angles are (4x + 15)° and (6x − 5)°.
Vertically opposite angles are equal: 4x + 15 = 6x − 5 15 + 5 = 6x − 4x 20 = 2x x = 10
Each angle: 4(10) + 15 = 55°. Check: 6(10) − 5 = 55° ✓
Alternate angles example: Two alternate angles (formed by a transversal crossing parallel lines) are (3x + 7)° and (5x − 9)°.
Alternate angles are equal: 3x + 7 = 5x − 9 7 + 9 = 5x − 3x 16 = 2x x = 8
Each angle: 3(8) + 7 = 31°. Check: 5(8) − 9 = 31° ✓
What is the full solution process?
- Identify the angle rule from the diagram description (straight line, triangle, quadrilateral, vertically opposite, parallel lines).
- Write the equation by setting the expression for angles equal to the appropriate sum (180°, 360°, or equal).
- Expand any brackets (if present).
- Collect like terms to a simple equation in x.
- Solve for x.
- Find each angle by substituting x back.
- Check that the angles satisfy the original rule.
Frequently asked questions
Can x represent a negative angle?
In a correctly formed diagram, all angles must be positive. If solving gives a negative angle, you have either made an algebraic error or the question has an error. Check your working — most commonly a sign has been lost when expanding or collecting terms.
What if there are two unknowns, x and y?
You need two independent equations — one from each angle rule that applies in the diagram. For example, a triangle with both parallel line constraints and an angle sum constraint gives two equations. Solve them simultaneously (using the substitution or elimination methods).
Why must I check my answer by substituting back?
Because the check tests both your algebra AND whether the angle rule you applied was appropriate. A check that does not give the expected sum (180°, 360° or equal) means either the wrong rule was used or the arithmetic contains an error. Finding this at the check stage is much better than submitting a wrong answer.
How does this topic link to GCSE proof?
At GCSE, algebraic angle questions often become proof questions: "Show that angle ABC = 90°." The method is the same — set up equations using angle facts, substitute known expressions, and show that the result equals the stated value. The difference is that in a proof you do not solve for x; instead you demonstrate the target result algebraically, often leaving the expression in terms of the given variables.
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