Synthetic Geometry and Constructions
Synthetic Geometry and Constructions is the deductive side of geometry: congruent and similar triangles, circle theorems, and straight-edge-and-compass constructions. You must quote the theorem or test you use and show construction arcs clearly.
Congruence and similarity
- Congruent triangles: SSS, SAS, ASA, RHS.
- Similar triangles: corresponding angles equal (AA) or corresponding sides in proportion; the scale factor relates lengths.
- In similar figures, areas scale by the square of the scale factor.
Circle theorems
- The angle at the centre is twice the angle at the circumference on the same arc.
- An angle in a semicircle is 90°.
- Opposite angles of a cyclic quadrilateral sum to 180°.
- A tangent is perpendicular to the radius at the point of contact.
Constructions
Know how to construct: the bisector of an angle; the perpendicular bisector of a segment; a perpendicular at a point on a line; a line parallel to a given line through a point; triangles given SSS, SAS or ASA. Keep compass arcs visible and label points.
Worked example
In a circle, an angle at the centre is 80°. Find the angle at the circumference standing on the same arc. By the centre-angle theorem, the angle at the circumference is half the angle at the centre: 80° ÷ 2 = 40°.
Traps
- Assuming a diagram is drawn to scale.
- Naming a theorem without applying it to the correct angle or arc.
- Rubbing out construction arcs that are needed as evidence.
- Confusing the tangent–radius theorem with the angle-in-semicircle theorem.