Angles

What is an Angle?

An angle is a measure of the amount of turn between two straight lines that meet at a point called the vertex. Angles are measured in degrees (°), where a full rotation equals 360°.

Angles appear everywhere in geometry — in shapes, parallel lines, bearings, and constructions. Understanding their properties and rules is essential for solving problems efficiently.

Types of Angles

Type	Size	Description
Acute	0° to 90°	Less than a right angle
Right	Exactly 90°	A perfect corner, marked with a small square
Obtuse	90° to 180°	Greater than a right angle but less than a straight line
Straight	Exactly 180°	A flat line
Reflex	180° to 360°	Greater than a straight angle
Full turn	Exactly 360°	A complete rotation

When identifying angle types, always compare to the 90° benchmark first: less than 90° is acute, greater is obtuse (up to 180°), then reflex.

Key Angle Rules

These rules let you find missing angles without measuring:

Angles on a straight line — add up to 180°

• If one angle is 65°, the other on the same line is 180° - 65° = 115°

Angles around a point — add up to 360°

• Four angles at a point: if three are 80°, 100°, 70°, the fourth is 360° - 250° = 110°

Vertically opposite angles — when two lines cross, the angles directly opposite each other are equal

• If one angle is 42°, its vertically opposite angle is also 42°

Complementary angles — two angles that add up to 90°

Supplementary angles — two angles that add up to 180°

Angles in Triangles

The interior angles of any triangle always add up to 180°.

Worked example: A triangle has angles 47° and 63°. Find the third angle.

• 47 + 63 = 110
• Third angle = 180 - 110 = 70°

The exterior angle of a triangle equals the sum of the two non-adjacent (remote) interior angles.

• If interior angles are 40° and 70°, the exterior angle is 40 + 70 = 110°

Special triangles:

• Equilateral — all angles are 60°
• Isosceles — two equal base angles
• Right-angled — one angle is exactly 90°

Angles in Polygons

For any polygon, the sum of interior angles = (n - 2) × 180°, where n is the number of sides.

Shape	Sides (n)	Interior angle sum
Triangle	3	180°
Quadrilateral	4	360°
Pentagon	5	540°
Hexagon	6	720°

For a regular polygon (all sides and angles equal), each interior angle = sum ÷ n.

Angles and Parallel Lines

When a transversal (a line crossing two parallel lines) is drawn, four important angle relationships are created. In the diagrams below, lines p and q are parallel:

Alternate angles — on opposite sides of the transversal, between the parallel lines. They are equal.

• Also called "Z angles" because of the Z shape they form.

Corresponding angles — on the same side of the transversal, one between the parallels and one outside. They are equal.

• Also called "F angles".

Co-interior angles (same-side interior) — on the same side of the transversal, between the parallel lines. They add up to 180°.

• Also called "C angles" or allied angles.

Vertically opposite angles — formed at each intersection, equal at every crossing point.

Worked Example: Parallel Lines

Two parallel lines are cut by a transversal. One angle is 55°. Find the alternate, corresponding, and co-interior angles.

• Alternate angle = 55° (equal)
• Corresponding angle = 55° (equal)
• Co-interior angle = 180° - 55° = 125°

Bearings

A bearing is a direction measured clockwise from north, always written as a three-digit number.

• North = 000°
• East = 090°
• South = 180°
• West = 270°

Example: A ship travels on a bearing of 065°. To find the back-bearing (return direction), add 180°: 065° + 180° = 245°.

Bearings use the same angle rules — alternate, corresponding, and co-interior angles often appear in bearing problems involving parallel north lines.

Common Mistakes

• Confusing obtuse and reflex — obtuse is between 90° and 180°; reflex is between 180° and 360°.
• Misidentifying alternate and co-interior — alternate angles are equal; co-interior angles add to 180°.
• Forgetting the polygon formula — many students guess 360° for all polygons; it is only correct for quadrilaterals.
• Not writing reasons — in exam questions, always state the angle rule you used (e.g. "angles on a straight line").
• Bearing errors — forgetting to measure clockwise from north, or writing a two-digit bearing instead of three digits.

Tips and Tricks

• Draw and label diagrams clearly — marking equal angles with the same arc symbol prevents confusion.
• When finding multiple missing angles, work step by step and label each answer before moving on.
• For parallel line problems, extend or redraw the transversal line to make the Z, F, or C shape visible.
• Always check: if you have found all angles at a point, do they sum to 360°? If not, recheck your work.