Properties and constructions

Angle questions are marked on the reason as much as the answer, and the reasons are a fixed vocabulary. Write the reason beside each step as you go — "alternate angles", "angles on a straight line", "base angles of an isosceles triangle" — because a correct number with no reason scores less than the full mark.

Angles in a polygon come from splitting it into triangles: a polygon with n sides splits into n − 2 triangles, so its interior angles sum to (n − 2) × 180°. Exterior angles always sum to 360° however many sides there are, which makes them the quicker route for a regular polygon: each exterior angle is 360° ÷ n, and the interior angle is 180° minus that.

Congruent shapes are identical in size and shape, though one may be rotated or reflected. Similar shapes have the same angles with all sides in the same ratio — one is an enlargement of the other.

To prove two triangles congruent, quote one of exactly four criteria and show the three facts it needs:

• SSS — all three sides equal • SAS — two sides and the angle between them • ASA — two angles and a corresponding side • RHS — right angle, hypotenuse and one other side

SSA is not a criterion and proves nothing. A proof must name the criterion, list the three matching parts with a reason for each, and state the conclusion.

Similarity questions use the scale factor: find it from a pair of corresponding sides, then multiply or divide. Corresponding sides must be matched correctly, which is easiest if the two shapes are redrawn the same way round.

Fractional and negative scale factors are Higher-tier requirements. A fractional scale factor between 0 and 1 makes the image smaller, and it is still called an enlargement. A negative scale factor puts the image on the opposite side of the centre, rotated through 180°.

To enlarge with a negative scale factor, measure from the centre to a point, then go the same multiple of that distance in the opposite direction.

Combinations of transformations are also examined, and often a pair of them is equivalent to a single one — two reflections in parallel mirror lines give a translation, and two reflections in perpendicular lines give a rotation of 180°. Questions ask you to describe that single equivalent transformation fully.

Invariance means what stays unchanged. Under translation, reflection and rotation the lengths and angles are invariant, so the shape is congruent. Under enlargement only the angles are invariant, so the shape is similar. An invariant point is one that does not move — the centre of a rotation or enlargement, and any point lying on the mirror line of a reflection.

Constructions are done with a ruler and compasses, and the construction arcs must be left visible — they are what the marks are for. Never measure with a protractor when a construction is asked for.

The three standard constructions:

• Perpendicular bisector of a line segment — open the compasses to more than half the length, draw arcs from both ends above and below the line, and join where they cross. Every point on it is equidistant from the two ends. • Perpendicular from or at a point on a line — arcs either side of the point on the line, then arcs from those two positions, and join. • Angle bisector — an arc from the vertex crossing both arms, then equal arcs from those two crossings, and join to the vertex. Every point on it is equidistant from the two arms.

The perpendicular distance from a point to a line is the shortest distance to that line, which is why loci questions about "nearest" use perpendiculars.

The circle vocabulary comes first, because the theorems are stated in it: the centre, a radius from centre to edge, a chord joining two points on the circumference, the diameter as a chord through the centre, the circumference as the perimeter, a tangent touching at one point, an arc as part of the circumference, a sector as a slice between two radii, and a segment as the region cut off by a chord.

The specification asks you to prove these as well as apply them, and the proofs use the angle facts from earlier in this sub-topic — chiefly that radii are equal, so any triangle with two radii is isosceles and its base angles are equal.

To prove that the angle at the centre is twice the angle at the circumference: draw a radius from the centre to the point on the circumference, making two isosceles triangles. Label the base angles of each, use the exterior angle of a triangle equals the sum of the two opposite interior angles on each triangle, and add the results. The angle at the centre comes out as twice the angle at the circumference.

When applying them, start by marking on the diagram every right angle, every equal radius and the centre. Most problems open up once those are visible.

Faces are the flat surfaces, edges are where two faces meet, and vertices are the corners. A cube has 6 faces, 12 edges and 8 vertices; a square-based pyramid has 5 faces, 8 edges and 5 vertices. A cylinder, cone and sphere have curved surfaces, so their face and edge counts are described rather than counted in the same way.

A prism has the same cross-section all the way through, which is what makes its volume the area of that cross-section multiplied by the length.

Plans and elevations are the views from three directions, drawn to scale on squared paper:

• The plan is the view from directly above • The front elevation is the view from the front • The side elevation is the view from the side

Each is a flat 2D drawing with no perspective, and internal edges where the depth changes are drawn as solid lines. Given three views you may also be asked to work backwards and describe or draw the solid.