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Geometry: Angles and Polygons Explained

Angles and polygons are the building blocks of flat geometry. Mastering a handful of angle rules — for straight lines, parallel lines, and the interior angles of polygons — unlocks the ability to solve a huge range of exam problems and provides the logical foundation for more advanced work in trigonometry and coordinate geometry.

What Is an Angle?

An angle is a measure of rotation between two rays that share a common endpoint called the vertex. Angles are measured in degrees (°), where a full rotation is 360°. They can also be measured in radians (used in higher-level mathematics), where a full rotation equals 2π radians.

Types of Angle

• Acute angle: greater than 0° and less than 90°. Example: 45°.
• Right angle: exactly 90°. Marked with a small square in diagrams.
• Obtuse angle: greater than 90° and less than 180°. Example: 130°.
• Straight angle: exactly 180° — a flat line.
• Reflex angle: greater than 180° and less than 360°. Example: 250°.
• Full turn: exactly 360°.

When two angles add up to 90° they are called complementary. When they add up to 180° they are called supplementary.

Angle Rules on Lines and Points

Several rules follow directly from the definition of angles and allow you to find unknown angles without measuring:

• Angles on a straight line sum to 180°. If two or more angles sit side by side on one side of a straight line and share the same vertex, they add up to 180°.
• Angles around a point sum to 360°. All angles meeting at a single point with no gaps together make a full revolution.
• Vertically opposite angles are equal. When two straight lines cross, the angles directly across from each other (not adjacent) are always equal. They are called vertically opposite because both angles share the same vertex and their arms form opposite rays. This rule is also known as the X-rule.

In an exam question, you can often chain these three rules together to find several unknown angles from just one given angle.

Angles and Parallel Lines

When a straight line (called a transversal) crosses two parallel lines, it creates eight angles. These angles group into three important pairs:

• Corresponding angles are equal. They sit on the same side of the transversal and on the same side of their respective parallel line — one at the top of an intersection, one at the top of the other. They form an F-shape (or backwards F). Remembered as: corresponding → equal.
• Alternate angles are equal. They sit on opposite sides of the transversal between the two parallel lines — one above the lower parallel line, one below the upper parallel line. They form a Z-shape (or S-shape). Remembered as: alternate → equal.
• Co-interior angles (also called same-side interior or allied angles) add up to 180°. They sit on the same side of the transversal between the two parallel lines. They form a C-shape (or U-shape). Remembered as: co-interior → supplementary.

These rules only hold when the lines are genuinely parallel. Always state which rule you used in your working — examiners award marks for reasoning, not just the answer.

Angles in Triangles

The three interior angles of any triangle always add up to 180°. This follows from the parallel-lines rules and can be proved by drawing a line through one vertex parallel to the opposite side.

An exterior angle of a triangle is formed by extending one side. The exterior angle equals the sum of the two non-adjacent interior angles. This is the exterior angle theorem: if one interior angle is 70° and another is 50°, the exterior angle at the third vertex is 70 + 50 = 120°.

Special triangles by angle:

• Equilateral: all three angles equal 60°; all sides equal.
• Isosceles: two equal sides; the two base angles (opposite the equal sides) are equal.
• Scalene: all three sides and angles different.
• Right-angled: one angle exactly 90°; the other two are complementary.

Interior Angles of Polygons

A polygon is a closed flat shape with straight sides. The sum of the interior angles of any polygon with n sides is given by:

Sum of interior angles = (n − 2) × 180°

This formula works because any polygon can be divided into triangles by drawing diagonals from one vertex, and each triangle contributes 180°. A quadrilateral (n = 4) splits into 2 triangles: (4 − 2) × 180 = 360°. A pentagon (n = 5): (5 − 2) × 180 = 540°. A hexagon (n = 6): (6 − 2) × 180 = 720°.

For a regular polygon (all sides and angles equal), each interior angle is:

Each interior angle = (n − 2) × 180° ÷ n

A regular hexagon: (6 − 2) × 180 ÷ 6 = 720 ÷ 6 = 120°. A regular octagon: (8 − 2) × 180 ÷ 8 = 1080 ÷ 8 = 135°.

Exterior Angles of Polygons

An exterior angle of a polygon is the angle between one side and the extension of the adjacent side. For any convex polygon, the exterior angles always add up to 360°, regardless of the number of sides. Think of it this way: if you walk all the way around the perimeter and turn at every corner, you end up facing the same direction you started — one complete rotation.

For a regular polygon, each exterior angle = 360° ÷ n. A regular pentagon: 360 ÷ 5 = 72°. Interior and exterior angles at any vertex are supplementary (they add to 180°), so a regular pentagon's interior angle is 180 − 72 = 108° — matching the formula above.

The exterior-angles-sum rule also lets you find the number of sides of a regular polygon if you know just one exterior angle: n = 360 ÷ (exterior angle). An exterior angle of 40° means n = 360 ÷ 40 = 9 sides (a nonagon).

Summary

Angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal. When a transversal crosses parallel lines, corresponding and alternate angles are equal, while co-interior angles sum to 180°. Triangle angles sum to 180° and an exterior angle equals the sum of the two non-adjacent interior angles. The interior angle sum of an n-sided polygon is (n − 2) × 180°; the exterior angles always sum to 360°. Mastering these rules — and being able to state the reason for each step — is the key to scoring full marks on geometry problems at every level.