Angles in Polygons & Parallel Lines

• How to correctly label line segments, angles, triangles, quadrilaterals and other polygons.
• The four "basic angle facts": angles around a point (360°), angles on a straight line (180°), vertically opposite angles (equal).
• Angle properties of triangles (equilateral, isosceles, scalene, right-angled).
• Angle properties of quadrilaterals (square, rectangle, parallelogram, rhombus, kite).
• Interior and exterior angles of any polygon, and the formulas that connect them to the number of sides, n.
• Finding a missing angle or missing number of sides in a regular polygon.
• Angles formed by a line crossing two parallel lines: corresponding (F), alternate (Z), and co-interior (C) angles.

Before you can talk about angles properly, you need to know how mathematicians name them — and examiners are strict about this, because a wrong label can mean a wrong answer even if your maths was right.

• The line segment AB simply means the straight bit connecting point A to point B.
• The angle ABC is the angle sitting at point B, formed where line segment AB meets line segment BC. Notice the middle letter (B) is always where the angle actually is — that's the rule.
• Unless told otherwise, "angle ABC" means the acute or obtuse angle at that corner — not the reflex angle (the "long way round").
• The triangle ABC is made of segments AB, BC, and CA.
• A quadrilateral ABCD is made of segments AB, BC, CD, and DA — the letters must be written in order around the shape (either clockwise or anticlockwise). Writing them out of order, like ABDC, describes a completely different (usually self-crossing) shape!

These three facts are the absolute foundation of the entire chapter. Almost every question — however complicated it looks — eventually comes down to one of these three ideas.

Whatever type of triangle you're given, one rule never changes:

On top of that base rule, different triangle types give you extra shortcuts:

• Square / Rectangle: all four angles are 90°.
• Parallelogram / Rhombus: opposite angles are equal (not all four — just the pairs facing each other).
• Kite: only one pair of opposite angles are equal (the pair between the unequal sides).

A polygon is just any 2D shape made entirely of straight sides — n of them, where n can be any whole number 3 or more (3 sides = triangle, 4 = quadrilateral, 5 = pentagon, and so on).

A regular polygon is the "perfectly symmetrical" version: every side is the same length, and every angle is the same size. A regular 3-sided polygon is an equilateral triangle; a regular 4-sided polygon is a square.

This trips a lot of students up, so let's be very precise:

• Interior angle: the angle inside the polygon, at a corner (vertex).
• Exterior angle: the angle you'd need to continue one side out past the corner to make a straight line with the interior angle. It is NOT the reflex angle that goes all the way around the outside of the corner — that's a very common misconception!

Here's the key insight: any polygon with n sides can be split into (n − 2) triangles by drawing diagonals from a single vertex. Since each triangle contributes 180°, we get:

Worth memorising directly (you'll use these constantly):

Because every angle in a regular polygon is identical, you can divide the totals above by n to get a single angle:

Finding the number of sides, given an angle

If you're told the exterior angle, just rearrange: n = 360 ÷ exterior angle.

If you're told the interior angle, set it equal to the formula and solve for n:

Parallel lines are lines that stay exactly the same distance apart forever — no matter how far you extend them in either direction, they will never touch or cross. When a third straight line (called a transversal) cuts across two parallel lines, it creates a set of predictable, nameable angle relationships. Learning to spot the shape the angles make is the whole trick to this topic.

Corresponding angles sit in matching positions at each intersection — think of them as "the same corner" at two different crossing points. Trace the shape they make and you'll see a letter F (sometimes flipped, rotated, or upside-down — don't be fooled if it doesn't look like a textbook F straight away).

Alternate angles sit on opposite sides of the transversal, between the two parallel lines. They trace out a Z shape (or a rotated/reflected "N" or "S" — same idea).

Co-interior angles sit on the same side of the transversal, between the parallel lines. They trace a C shape (or backwards C / U shape).

In a real exam question, you'll often need to combine vertically opposite angles with the F/Z/C rules. A brilliant way to see this: once you know one angle (call it x°) at a crossing point, you instantly know THREE more angles at that same point:

Once you have this "map" of angles at ONE crossing point, the F, Z, and C rules let you transfer that same information across to the SECOND crossing point on the other parallel line — meaning you often know 8 angles total from just 1 given value!