Angles, polygons, congruence and vectors

Angles and polygons (M5.1-M5.3)

• Angles on a line \(180^\circ\), at a point \(360^\circ\), vertically opposite equal; with parallel lines: alternate equal, corresponding equal, co-interior sum to \(180^\circ\).
• Interior angles of an \(n\)-gon sum to \(180(n - 2)^\circ\); exterior angles sum to \(360^\circ\); regular \(n\)-gon exterior angle \(\frac{360}{n}\).
• Quadrilaterals: parallelogram (opposite sides parallel and equal), rhombus (parallelogram with equal sides, perpendicular diagonals), rectangle (equal diagonals), square (both), kite (perpendicular diagonals, one line of symmetry), trapezium (one pair of parallel sides).

Congruence, similarity, transformations, vectors (M5.4-M5.6, M5.14, M5.18, M5.19)

• Congruence: SSS, SAS, ASA (or AAS), RHS. Two sides and a non-included angle (SSA) is not enough.
• Similar shapes: equal angles, sides in the same ratio; a line parallel to one side of a triangle creates a similar triangle.
• Transformations: enlargement with negative scale factor inverts through the centre; rotation \(90^\circ\) anticlockwise about \(O\): \((x, y) \to (-y, x)\); reflection in \(y = x\): \((x, y) \to (y, x)\).
• Bearings: measured clockwise from north, three figures; the back bearing differs by \(180^\circ\).
• Vectors: \(\overrightarrow{AB} = \mathbf b - \mathbf a\); parallel vectors are multiples; collinear points have parallel joining vectors; midpoint \(\frac12(\mathbf a + \mathbf b)\); ratio point \(OP : PM = 2 : 1\) gives \(\overrightarrow{OP} = \frac23\overrightarrow{OM}\).