Geometry – Properties of Shape

Triangles are classified on two independent systems: by angle (acute, right, obtuse) and by side (equilateral, isosceles, scalene). A full classification needs both labels — 'right isosceles' is a complete, commonly tested answer. Check the two systems separately, since a triangle can be obtuse and scalene, or right and isosceles, but equilateral only ever pairs with acute.

The interior angle sum of any convex $n$-gon is $(n-2)\times 180°$, found by splitting the polygon into $(n-2)$ triangles from one vertex. Exterior angles of any convex polygon always sum to $360°$, regardless of the number of sides. For a regular $n$-gon, each interior angle equals $\frac{(n-2)\times 180°}{n}$.

To classify a triangle as acute, right, or obtuse from side lengths alone, compare $a^2+b^2$ with $c^2$, where $c$ is the longest side. Mark schemes reward showing both numbers side by side (e.g. '50 vs 49') rather than a bare final answer. Real measurements carry tolerance, and that tolerance can genuinely shift which category a triangle falls into — check whether $\sqrt{a^2+b^2}$ lands inside the stated error range before claiming certainty.

Line symmetry counts the fold lines that map a shape onto itself; rotational symmetry counts how many times a shape matches itself within one full turn, called its order. A regular $n$-gon always has $n$ lines of symmetry and rotational symmetry of order $n$. A shape tessellates alone only if its interior angle divides $360°$ exactly — true for equilateral triangles, squares, and regular hexagons, but false for regular pentagons.

Congruent shapes match exactly in every side and angle, proven with minimum tests: SSS, SAS, ASA, AAS, and RHS (right triangles only). Similar shapes have equal corresponding angles and sides in a fixed ratio, the scale factor $k$, proven with AA, SAS, or SSS similarity tests. Scale factor stretches length by $k$, but area scales by $k^2$ and volume by $k^3$, since area and volume are two- and three-dimensional measurements.