📋 Summary — What This Chapter Covers
- 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.
1️⃣ Basic Angle Properties
Labelling lines, angles and shapes
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!
A
/\
/ \
/ \
/______\
C B
Triangle ABC — made of
segments AB, BC, and CA.
The angle "ACB" sits at point C.
Quick check
For a quadrilateral labelled A → B → C → D, going around in order (clockwise or anticlockwise) keeps the shape correct. ABDC breaks that order and gives you a totally different, crossed shape — always trace the letters with your finger to check!
The three fundamental angle facts
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.
Angles around a full point = 360°
Think of spinning all the way around and ending up facing the same way — that's one full turn, 360°.
Angles on a straight line = 180°
A straight line is "half a turn" — so any angles that sit along it and fill it completely must add to 180°.
Vertically opposite angles are equal
When two straight lines cross, they form an "X" shape. The two angles directly opposite each other across the crossing point are always identical.
\ /
\ /
---X--- Vertically opposite:
/ \ the top and bottom angles are equal.
/ \ The left and right angles are equal
(but a different value from top/bottom).
Angle properties of triangles
Whatever type of triangle you're given, one rule never changes:
Interior angles of any triangle sum to 180°
On top of that base rule, different triangle types give you extra shortcuts: