Everything in this topic comes from three facts: angles on a straight line make 180°, a triangle makes 180°, and parallel lines copy their angles across.
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Give the angles names and the question answers itself.
📏 The angle words
| Name | Meaning |
|---|---|
| Complementary | two angles adding to 90° |
| Supplementary | two angles adding to 180° |
| Vertically opposite | the pair across a crossing — always equal |
| Corresponding | same position at each parallel line — equal |
| Alternate | opposite sides of the transversal, inside — equal |
| Co-interior | same side of the transversal, inside — supplementary |
Co-interior angles are the odd one out. Corresponding and alternate angles are equal; co-interior angles add to 180°. Getting these the wrong way round costs marks in every paper.
The 180° rule and the exterior-angle rule between them solve almost everything.
🔺 Triangle facts
The three angles add up to 180°.
An exterior angle equals the sum of the two interior opposite angles. So a triangle with angles 50° and 60° has an exterior angle of 110° at the third vertex.
The sum of any two sides is greater than the third — so 3, 4, 8 cannot be a triangle at all.
In an isosceles triangle the angles opposite the equal sides are equal.
📐 Congruence rules
| Rule | What must match |
|---|---|
| SSS | all three sides |
| SAS | two sides and the angle between them |
| ASA | two angles and the side between them |
| RHS | right angle, hypotenuse and one side |
AAA is not a congruence rule. Two triangles can have identical angles and completely different sizes — they are similar, not congruent. This is a standard trap option.
Two kinds: reflection (lines of symmetry) and rotation (order of symmetry).
🪞 Counted out
| Shape | Lines of symmetry | Order of rotation |
|---|---|---|
| Equilateral triangle | 3 | 3 |
| Square | 4 | 4 |
| Rectangle | 2 | 2 |
| Rhombus | 2 | 2 |
| Regular pentagon | 5 | 5 |
| Parallelogram | 0 | 2 |
| Circle | infinitely many | infinite |
A regular polygon of n sides has n lines of symmetry and rotational symmetry of order n. That one sentence answers most symmetry questions in the paper.
A parallelogram has no line of symmetry at all, even though it has rotational symmetry of order 2. Its diagonals are not lines of symmetry — fold it and the halves do not match.
12 Olympiad questions on angles, triangles and symmetry. 🍀
