Maths reference: Angles and shapes

Parallel lines

\text{parallel lines} \gg \text{ crossed by a transversal}

transversal

Parallel lines go the same way and never meet; arrows (>>) mark them. A transversal crosses both, making eight angles.

a = e

corresponding angles

The same place at each crossing (an F shape). With parallel lines they are equal.

a and e: the same place at each crossing, both 120°

c = f

alternate angles

Between the parallel lines, on opposite sides of the transversal (a Z shape). With parallel lines they are equal.

c and f: between the lines, opposite sides, both 60°

c + e = 180^\circ

co-interior angles

Between the parallel lines, on the same side (a C shape). With parallel lines they add to $180^\circ$ : they are not equal unless both are $90^\circ$ .

c and e: between the lines, same side, 60° + 120° = 180°

b = c

vertically opposite angles

Opposite each other at one crossing: always equal, parallel lines or not. Not one of the parallel-line pairs.

f = 70^\circ \;\text{(corresponding to } b\text{)}

give a reason

For each angle you find, name the rule: corresponding or alternate angles are equal; co-interior angles add to $180^\circ$ ; angles on a straight line add to $180^\circ$ .

Words

\angle A

interior angle

An angle inside a shape, at one of its corners.

n \text{ sides}, \; n \text{ corners}

polygon

A closed shape with straight sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), octagon (8).

\text{each angle} = \dfrac{(n - 2) \times 180^\circ}{n}

regular polygon

All sides and all angles are equal, so each angle is the angle sum shared equally: a regular hexagon has angles of $120^\circ$ .

Rules

a + b + c = 180^\circ

angle sum of a triangle

True for every triangle, whatever its shape or size: tear off the corners and they fill a straight line.

An obtuse triangle: 24° + 127° + 29° = 180°

(n - 2) \times 180^\circ

angle sum of a polygon

Split it into triangles from one corner: a shape with $n$ sides makes $n - 2$ triangles. A pentagon: $3 \times 180^\circ = 540^\circ$ .

A pentagon splits into 3 triangles from A: 540°

Shapes

\text{scalene, isosceles, equilateral} \;\; \text{acute, right-angled, obtuse}

triangles by sides and angles

Give two names: one for the sides (how many are equal) and one for the biggest angle. Equal sides sit opposite equal angles.

An isosceles acute triangle, its equal sides ticked

\text{every square is a rectangle; every rectangle is a parallelogram}

quadrilateral family

Trapezium: at least one pair of parallel sides. Parallelogram: two. Rhombus: four equal sides. Rectangle: four right angles. Square: both. Kite: two pairs of equal neighbouring sides. A square is also a rectangle and a rhombus.

A square with equal sides, parallel sides and right angles marked

\text{equal sides} \;\text{and}\; \text{equal angles}

regular polygon

Both are needed: a rhombus has equal sides but is not regular. Names: pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10.

\text{concave: an angle} > 180^\circ

convex and concave

Convex: every corner points out. Concave: at least one corner points in.

A concave hexagon with one corner pointing in

Solids, transformations and algorithms

\text{cube: 6 squares on 6 different faces}

net

A net is a solid unfolded flat. Fold it, or roll a cube over it, to check that every face is covered once. Six squares is not enough.

A cross of six squares that folds into a cube

A = 2 \times \text{triangle} + \text{length} \times (a + b + c)

prism net area

A triangular prism has two triangles and three rectangles. The net's area is the surface area.

\text{front} = \text{tallest pile in each column}

top, front and side views

A view shows what you can see from one side; hidden cubes do not show. The top view with heights shows every cube.

(x, y) \to (x + a, \, y + b)

translation

Every point slides the same way. Right and up add; left and down take away.

x\text{-axis: } (x, y) \to (x, -y) \qquad y\text{-axis: } (x, y) \to (-x, y)

reflection

Each point and its image are the same distance from the mirror, on opposite sides.

90^\circ \text{ anticlockwise: } (x, y) \to (-y, x)

rotation

Give the centre, the angle and the direction. A half turn: (x, y) to (−x, −y). 90° clockwise is 270° anticlockwise.

A triangle turned a quarter turn about the origin

\text{ask the most specific question first}

algorithm

A set of steps and decisions that gives the right result every time. Test it on one shape of every class.

Congruence, similarity and 3D

\text{SSS},\ \text{SAS},\ \text{ASA or AAS},\ \text{RHS}

congruence tests

Each set of equal parts fixes a triangle. In SAS the angle is between the two sides; SSA is not a test, because it can give two triangles.

Two different triangles with the same SSA parts

k = \dfrac{\text{copy length}}{\text{matching original length}}

scale factor

Similar shapes: every length times k, every angle the same. Triangles are similar by AA, SSS (sides in one ratio) or SAS.

A 3, 4, 5 triangle and a copy twice as big

\text{rectangle: equal, bisect} \quad \text{rhombus: bisect at } 90^\circ

diagonals of quadrilaterals

A parallelogram's diagonals cut each other in half; a kite's cross at 90°. Prove a property with two congruent triangles.

A rectangle with its two equal diagonals

\text{opposite angles equal; neighbours add to } 180^\circ

parallelogram angles

Neighbouring angles are co-interior between parallel sides. All four add to 360°.

(x, y, z)

3D coordinates

Along x, along y, then up z. The order is always x, y, z; points on the floor have z = 0.

\text{congruent?} \to \text{similar?} \to \text{not enough to tell}

congruence algorithm

Ask the congruence tests first, then the similarity conditions, and end with a catch-all. Test on one pair of each kind, including SSA.