Using the angle properties of triangles, quadrilaterals and polygons, angles in parallel lines, and the symmetry and angle properties of the circle to calculate missing angles.

What this dot point is asking

The SQA wants you to use angle facts to calculate missing angles in triangles, quadrilaterals and polygons, in parallel-line diagrams, and in circles, justifying each step with the property you used. Clear reasoning matters as much as the final number.

Angles in triangles and polygons

The angle sum of a triangle is always $180^\circ$, and of a quadrilateral $360^\circ$. For any polygon, split it into triangles from one vertex.

Angles in parallel lines

When a straight line crosses a pair of parallel lines, several equal and supplementary angles appear.

To solve a parallel-line problem, name the angle you can find, state the property, and chain the facts until you reach the required angle.

Angles in the circle

The circle has its own angle properties, several arising from its symmetry. Two radii to the ends of a chord form an isosceles triangle, which lets you find base angles.

Examples in context

Angle properties underpin construction and design. A roof truss is a triangle whose angles must sum to $180^\circ$, so a designer can find the third angle from the other two. Tiling patterns rely on regular polygons whose interior angles fit exactly around a point ($360^\circ$). Engineers use the tangent-radius right angle when a belt leaves a pulley, since the belt is tangent to the wheel.

Try this

Q1. Find the missing angle in a triangle with angles $40^\circ$ and $75^\circ$. [1 mark]

• Cue. $180 - 40 - 75 = 65^\circ$.

Q2. Find each interior angle of a regular octagon. [2 marks]

• Cue. $\dfrac{(8 - 2) \times 180}{8} = 135^\circ$.

Q3. $AB$ is a diameter and $C$ is on the circle. Angle $ABC = 60^\circ$. Find angle $BAC$. [2 marks]

• Cue. Angle $ACB = 90^\circ$, so angle $BAC = 30^\circ$.