Topic 4: Geometry

Topic 4 ties together naming, reasoning and accurate drawing. Most marks come from chaining a few standard facts: angles on a straight line add to a fixed total, parallel lines create equal and supplementary pairs, and every polygon has predictable interior and exterior angle sums. Circle theorems add a powerful layer for any diagram containing a circle, while similarity and congruence let you transfer information between shapes using scale factors. The drawing skills, constructions, nets and bearings, demand neat work with a sharp pencil, ruler, protractor and pair of compasses. Throughout, always state the reason for each step: examiners award method marks for correctly named angle facts even when the final number slips.

Geometrical terms and naming

Lines and angles have exact names you must use. An acute angle is less than $90^{\circ}$, a right angle is exactly $90^{\circ}$, an obtuse angle is between $90^{\circ}$ and $180^{\circ}$, a straight angle is $180^{\circ}$, and a reflex angle is between $180^{\circ}$ and $360^{\circ}$. Two lines that meet at $90^{\circ}$ are perpendicular; lines that never meet and stay the same distance apart are parallel, usually marked with matching arrowheads. Triangles are classified by sides and angles. An equilateral triangle has three equal sides and three $60^{\circ}$ angles. An isosceles triangle has two equal sides and two equal base angles. A scalene triangle has all sides and angles different. A right-angled triangle contains one $90^{\circ}$ angle. Quadrilaterals also have set definitions: a square (four equal sides, four right angles), a rectangle (opposite sides equal, four right angles), a rhombus (four equal sides, opposite angles equal), a parallelogram (opposite sides parallel and equal), a trapezium (one pair of parallel sides), and a kite (two pairs of adjacent equal sides). Knowing which properties belong to which shape lets you deduce missing lengths and angles quickly.

Angles on lines and at a point

Three core facts solve most basic angle questions. Angles on a straight line add up to $180^{\circ}$, so if one angle is $115^{\circ}$ the other is $180^{\circ}-115^{\circ}=65^{\circ}$. Angles around a full point add up to $360^{\circ}$. Vertically opposite angles, formed where two straight lines cross, are equal. When writing solutions, name the rule. For example: 'angles on a straight line sum to $180^{\circ}$' or 'vertically opposite angles are equal'. This earns method marks. A common multi-step problem gives several angles meeting at a point and asks for the missing one: add the known angles and subtract from $360^{\circ}$. If three angles around a point are $90^{\circ}$, $140^{\circ}$ and $x$, then $x=360^{\circ}-90^{\circ}-140^{\circ}=130^{\circ}$.

Parallel lines

When a straight line, called a transversal, crosses two parallel lines, three equal-or-supplementary relationships appear. Corresponding angles (in matching positions, forming an F-shape) are equal. Alternate angles (on opposite sides of the transversal between the parallels, forming a Z-shape) are equal. Co-interior or allied angles (on the same side between the parallels, forming a C-shape) are supplementary, adding to $180^{\circ}$. Use the shape cues but always quote the proper name in your reasoning. For instance, if a transversal gives an angle of $72^{\circ}$, the alternate angle on the far parallel line is also $72^{\circ}$, and the co-interior angle is $180^{\circ}-72^{\circ}=108^{\circ}$. Many exam diagrams combine parallel-line facts with triangle facts, so identify all parallel pairs first, then transfer equal angles into the triangle to finish.

Triangles and polygons

The interior angles of any triangle sum to $180^{\circ}$. The exterior angle of a triangle equals the sum of the two interior opposite angles; this is a fast shortcut worth memorising. For an isosceles triangle, mark the two equal base angles equal and use the sum to find them. For a polygon with $n$ sides, the interior angles sum to $(n-2)\times 180^{\circ}$. The exterior angles of any convex polygon always sum to $360^{\circ}$. For a regular polygon (all sides and angles equal), each exterior angle $=\frac{360^{\circ}}{n}$, and each interior angle $=180^{\circ}-\frac{360^{\circ}}{n}$. These formulas let you work backwards. If a regular polygon has an exterior angle of $24^{\circ}$, then $n=\frac{360^{\circ}}{24^{\circ}}=15$ sides. If you know the interior angle is $156^{\circ}$, the exterior angle is $180^{\circ}-156^{\circ}=24^{\circ}$, again giving $15$ sides.

Circle theorems

Several circle theorems are examinable and recur constantly. The angle at the centre is twice the angle at the circumference standing on the same arc, so a central angle of $120^{\circ}$ gives a circumference angle of $60^{\circ}$. The angle in a semicircle is $90^{\circ}$ (a special case where the central angle is the $180^{\circ}$ diameter). Angles in the same segment, subtended by the same arc, are equal. A cyclic quadrilateral has all four vertices on the circle; its opposite angles are supplementary, summing to $180^{\circ}$. A tangent meets a radius at the point of contact at $90^{\circ}$. Two tangents drawn from the same external point are equal in length, creating an isosceles configuration. The alternate segment theorem states the angle between a tangent and a chord equals the angle in the alternate segment. When tackling a circle problem, label the centre, mark radii as equal (forming isosceles triangles), and note the diameter. State each theorem by name as you apply it. For example, if a diameter subtends a triangle at the circumference and one other angle is $35^{\circ}$, then the third angle is $180^{\circ}-90^{\circ}-35^{\circ}=55^{\circ}$, using the angle in a semicircle is $90^{\circ}$.

Congruence and similarity with scale factors

Two shapes are congruent if they are identical in size and shape; corresponding sides and angles are equal. The triangle congruence conditions are SSS (three sides), SAS (two sides and the included angle), ASA (two angles and a side), and RHS (right angle, hypotenuse and one side). To prove congruence, identify and state which condition applies. Two shapes are similar if they have the same shape but different size; corresponding angles are equal and corresponding sides are in the same ratio, called the linear scale factor $k$. To find a missing length, set up the ratio of matching sides. If a small triangle has a side of $4$ matched to $10$ in the larger triangle, then $k=\frac{10}{4}=2.5$, and every other length multiplies by $2.5$. Scale factors extend to area and volume. If the linear scale factor is $k$, the area scale factor is $k^{2}$ and the volume scale factor is $k^{3}$. So if two similar solids have a length ratio of $k=3$, their surface areas are in ratio $9$ and their volumes in ratio $27$. Working backwards, if volumes are in ratio $64$, then $k=\sqrt[3]{64}=4$ and areas are in ratio $16$.

Symmetry, constructions, nets and bearings

A 2D shape has line symmetry if a mirror line maps it onto itself; a square has $4$ lines of symmetry, a rectangle $2$, an equilateral triangle $3$. Rotational symmetry order is the number of times a shape looks the same in one full turn: a square has rotational symmetry of order $4$. For constructions, use a sharp pencil and compasses and leave all arcs visible. A perpendicular bisector of a line segment is drawn by opening the compasses to more than half the length and drawing arcs from each end above and below, then joining the crossings; every point on it is equidistant from the two ends. An angle bisector is made by drawing an arc from the vertex to cut both arms, then arcs from those two points, and joining the vertex to where they cross. A net is a 2D pattern that folds into a 3D solid. A cube net has six squares; a square-based pyramid net has a square with four triangles on its sides. Bearings are measured clockwise from north and always written with three figures, so due east is $090^{\circ}$ and south-west is $225^{\circ}$. To find a back bearing (the reverse direction), add $180^{\circ}$ if the bearing is less than $180^{\circ}$, or subtract $180^{\circ}$ if it is more. For example, the back bearing of $070^{\circ}$ is $070^{\circ}+180^{\circ}=250^{\circ}$, and the back bearing of $250^{\circ}$ is $250^{\circ}-180^{\circ}=070^{\circ}$.