Geometry: Triangles, Quadrilaterals, Congruence and Similarity
What you must be able to do
Know the angle facts: angles on a straight line sum to 180°, angles around a point to 360°, vertically opposite angles are equal, and alternate and corresponding angles formed by a transversal of parallel lines are equal.
For triangles: the interior angles sum to 180°; an exterior angle equals the sum of the two remote interior angles; the longest side faces the largest angle. Congruence criteria are SSS, SAS, ASA (with AAS) and the right-angle case HL. Similar triangles have equal angles and proportional sides; the ratio of areas equals the square of the similarity ratio.
For quadrilaterals: the interior angles sum to 360°. Know the properties of parallelograms (opposite sides and angles equal, diagonals bisect each other), rhombuses, rectangles, squares and trapezoids, and the midline theorem for triangles (the segment joining midpoints is parallel to the third side and half its length).
Traps that appear in questions
- Claiming AAA proves congruence - it proves only similarity.
- Assuming SSA is a congruence criterion (it is not, in general).
- Mixing up the similarity ratio with the area ratio.
- Using the exterior-angle theorem with the adjacent interior angle instead of the two remote ones.
- Assuming a quadrilateral with equal diagonals must be a rectangle without justification.
Rule of thumb
Mark every given equality on the figure before writing a proof, and name the criterion you use explicitly.
Worked example
Two similar triangles have sides in ratio 2 : 3. If the smaller has area 20 cm², the larger has area 20 · (3/2)² = 45 cm².