Every angle rule you need — from basic angle types through parallel lines, triangles, and polygons — explained with the reasoning examiners want to see.
An angle is a measure of the amount of turn between two straight lines that meet at a point. Angles are measured in degrees (°). A full turn is 360°.
The point where the two lines meet is called the vertex. The two lines are the arms of the angle.
These five rules appear in almost every geometry question. Know them by name — examiners award a mark for stating the correct reason in addition to the correct number.
Angles that form a straight line together add up to 180°.
Example: Two angles on a straight line are 65° and x. Find x.
x = 180° − 65° = 115°
Reason: angles on a straight line sum to 180°.
All angles around a single point add up to 360°.
Example: Three angles meet at a point: 120°, 95°, and y. Find y.
y = 360° − 120° − 95° = 145°
Reason: angles around a point sum to 360°.
When two straight lines cross, the angles opposite each other are equal. These are called vertically opposite angles.
If one angle is 72°, the angle directly opposite it is also 72°. The two remaining angles are both 108° (since 180° − 72° = 108°).
Reason: vertically opposite angles are equal.
The three interior angles of any triangle always add up to 180°.
Example: A triangle has angles 47°, 83°, and z. Find z.
z = 180° − 47° − 83° = 50°
Reason: angles in a triangle sum to 180°.
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles (the two angles that are not next to it).
Example: Two interior angles of a triangle are 40° and 65°. Find the exterior angle at the third vertex.
Exterior angle = 40° + 65° = 105°
Reason: exterior angle of a triangle = sum of the two opposite interior angles.
Worked example — isosceles triangle: An isosceles triangle has a top angle of 40°. Find the two base angles.
Remaining angles = 180° − 40° = 140°. Base angles are equal, so each = 140° ÷ 2 = 70°.
When a straight line (called a transversal) crosses two parallel lines, three important angle relationships are created. You must know all three names — examiners require the reason, not just the answer.
Corresponding angles are in the same position at each intersection — both above the parallel line, or both to the right of the transversal. They are equal.
Memory aid: they form an F-shape (or a backwards F).
Reason: corresponding angles are equal (parallel lines).
Alternate angles are on opposite sides of the transversal, between the two parallel lines. They are equal.
Memory aid: they form a Z-shape (sometimes called Z-angles).
Reason: alternate angles are equal (parallel lines).
Co-interior angles are on the same side of the transversal, between the two parallel lines. They add up to 180°.
Memory aid: they form a C-shape.
Reason: co-interior angles sum to 180° (parallel lines).
Two parallel lines are cut by a transversal. One angle is 55°. Find the alternate angle and the co-interior angle on the same side.
Alternate angle = 55° (alternate angles are equal).
Co-interior angle = 180° − 55° = 125° (co-interior angles sum to 180°).
A polygon is any flat shape with straight sides. The interior angles are the angles inside the shape.
For any polygon with n sides:
Sum of interior angles = (n − 2) × 180°
A regular polygon has all sides equal and all interior angles equal. Divide the sum of interior angles by the number of sides.
Example: Find the interior angle of a regular octagon (8 sides).
Sum = (8 − 2) × 180° = 1080°
Each interior angle = 1080° ÷ 8 = 135°
An exterior angle is formed by extending one side of the polygon. For any polygon, the sum of all exterior angles is always 360°, regardless of the number of sides.
For a regular polygon: each exterior angle = 360° ÷ n.
Example: Find the exterior angle of a regular hexagon.
360° ÷ 6 = 60°
Interior angle + exterior angle = 180° (they form a straight line). So the interior angle of the hexagon = 180° − 60° = 120°. This is a useful cross-check.
Question: A regular polygon has an interior angle of 140°. How many sides does it have?
Exterior angle = 180° − 140° = 40°
Number of sides = 360° ÷ 40° = 9 sides (a nonagon)
Geometry is built on a small number of rules applied repeatedly. Master the seven rules above, always write a reason with each step, and angle problems become some of the most reliable marks on the paper.