Angles on Lines and at Points

La méthode, étape par étape

1. 1

   Angles on a straight line total 180°

   one angle 115° → the other is 180 − 115 = 65°

   A straight line is half a full turn. This is the most used rule in the topic and it applies wherever a line is crossed by another, whether or not the diagram makes the straight line obvious.

2. 2

   Angles at a point total 360°

   120° + 90° + x = 360° → x = 150°

   A full turn is 360 degrees, so angles meeting at a single point must account for all of it. Diagrams often show three or four angles meeting where only some are labelled, and this rule fills the gap.

3. 3

   Vertically opposite angles are equal

   two lines crossing → opposite angles match

   When two straight lines cross, the angles facing each other are equal. This follows from the straight-line rule applied twice, so it is a consequence rather than a separate fact — which is worth saying, because it makes it memorable.

4. 4

   With parallel lines, look for F, Z and C shapes

   corresponding (F) equal · alternate (Z) equal · co-interior (C) sum 180°

   Tracing the shape the two angles form with the parallel lines identifies the relationship. The letters are informal but reliable, and they give a student something to search a diagram for rather than staring at it.

5. 5

   Give a reason for every step

   x = 65° (angles on a straight line add to 180°)

   Angle questions award marks for reasoning as well as answers, and an unexplained correct number frequently scores less than a fully justified one. Naming the rule used costs a few words and is often worth a mark by itself.

Four rules cover almost everything

Straight line 180, point 360, vertically opposite equal, and the three parallel-line relationships. Nearly every angle question at this level is a chain of these applied one after another, and the difficulty is never the individual step but seeing which one applies where.

The practical method is to fill in every angle you can, not only the one asked for. Working outward from what is known usually reaches the target in two or three moves, whereas trying to jump straight to it tends to stall.

• Straight line → 180°
• Around a point → 360°
• Vertically opposite → equal
• Corresponding (F) → equal
• Alternate (Z) → equal
• Co-interior (C) → add to 180°

Co-interior is the one that catches people

Corresponding and alternate angles are equal; co-interior angles are not — they add to 180. Because two of the three parallel-line rules give equality, students assume the third does too, and the error is invisible unless the numbers happen to expose it.

The C shape is the cue: the two angles sit on the same side of the transversal, between the parallels, and together they span a straight line's worth of turn. Checking whether the angles are on the same side or opposite sides decides between equal and supplementary.

Why the reason is worth as much as the answer

Exam mark schemes for angle problems typically allocate marks for the correct value and for the correct justification. A student who writes only the number has done the harder part and left the easier mark on the table.

The reasons should name the rule in full — 'alternate angles are equal' rather than 'Z angles' — since informal names are not always accepted. It is a small habit that pays reliably.

How we teach angle rules

We require a written reason on every line of working from the first lesson, not just before exams. Students who have been allowed to write bare numbers for a year find the reasoning genuinely difficult to add later, because they have never had to articulate why a step was valid.

We also teach students to annotate diagrams heavily — marking parallel arrows, filling in every derivable angle. Angle problems are far more often solved by a well-marked diagram than by cleverness.