Angles.
Use angle sums and parallel-line relationships, with a clear geometrical reason at every step.
Use angle notation and angle sums · identify parallel-line angles · calculate polygon angles
Name the angle at the middle letter
An angle measures a turn. A quarter-turn is 90°, a half-turn is 180° and a full turn is 360°. These reference turns explain many angle sums before any formula is introduced.
At a straight line, adjacent angles together make a half-turn. Around one point, all the angles together complete a full turn. When using three-letter angle notation, the middle letter names the vertex, helping you identify exactly which turn is being measured.
In angle ABC, the vertex is B and the arms are BA and BC. Use a property as the reason for each step, not simply “it looks equal”.
| At a point | Angles sum to 360°. |
|---|---|
| On a straight line | Adjacent angles sum to 180°. |
| Vertically opposite | Opposite angles formed by two intersecting lines are equal. |
| Triangle | Interior angles sum to 180°. |
| Quadrilateral | Interior angles sum to 360°. |
Two angles of a triangle are 48° and 67°. Find the third.
Reason: angle sum of a triangle is 180°.
The vertex angle of an isosceles triangle is 38°. Find each base angle.
Reasons: base angles of an isosceles triangle are equal, and triangle angle sum is 180°.
Pause and explainWhere is the vertex of angle ABC?
At B, the middle letter. The rays run from B towards A and C.
Parallel lines: location determines the reason
A line crossing two parallel lines creates repeated angle relationships. The parallel lines keep the same direction, so matching turns made with the crossing line repeat.
Identify the transversal and the two parallel lines before naming corresponding, alternate or co-interior angles. Corresponding and alternate pairs are equal; co-interior pairs sum to 180°. These conclusions rely on parallelism, so point to the arrows or statement that establishes it.
Corresponding angles are equal: same relative position at each intersection.
Pause and explainCan you use alternate-angle equality when the two lines are not known to be parallel?
Not just from their appearance. The parallel condition must be given or proved.
Polygon angles: separate totals from individual angles
Draw diagonals from one vertex of a polygon. A quadrilateral becomes two triangles, a pentagon three, and the pattern continues. Since every triangle has angle sum 180°, the polygon formula counts those triangles.
For a regular polygon, the interior angles are equal, so divide the total by the number of sides. Exterior turning angles instead describe the total turn while walking around the polygon; one full circuit turns through 360°.
Draw diagonals from one vertex of a convex n-sided polygon. It divides into n − 2 triangles.
The exterior turning angles of a convex polygon sum to 360°. An interior angle and its adjacent exterior angle on a straight line sum to 180°.
Find the interior and exterior angles of a regular octagon.
Interior sum = (8 − 2) × 180° = 1080°. Each interior angle = 1080° ÷ 8 = 135°.
Each exterior angle = 360° ÷ 8 = 45°. Check: 135° + 45° = 180°.
A regular polygon has exterior angle 24°. How many sides?
Pause and explainWhy does a hexagon have interior sum 720°?
It can be split from one vertex into four triangles, giving 4 × 180° = 720°.
Your turn. Give the reason.
Mark the required angle and write one known relationship at a time. Include the reason, such as “angles on a straight line” or “alternate angles”.
Do not apply a regular-polygon formula unless equal angles are justified. Check whether the question asks for an interior angle, an exterior angle or the full angle sum.
Three angles around a point are 95°, 130° and x°. Find x.
Reveal worked answer
Co-interior angles between parallel lines are 3x° and (2x + 30)°. Find x.
Reveal worked answer
Find each interior angle of a regular decagon.
Reveal worked answer
Five angles of an irregular hexagon are 100°, 120°, 130°, 110° and 140°. Find the sixth.