Angles and shapes: recap

A line crossing two parallel lines (a transversal) makes eight angles, $a$ to $h$. Each pair below takes one angle from each crossing:

• corresponding: the same place at each crossing, such as $a$ and $e$ (an F shape);
• alternate: between the parallel lines, on opposite sides of the transversal, such as $c$ and $f$ (a Z shape);
• co-interior: between the parallel lines, on the same side, such as $c$ and $e$ (a C shape).

Two angles at one crossing, opposite each other, are vertically opposite: not one of these pairs.

When the lines are parallel:

• corresponding angles are equal;
• alternate angles are equal;
• co-interior angles add to $180^\circ$ (they are not equal unless both are $90^\circ$).

With vertically opposite angles (equal) and angles on a straight line (add to $180^\circ$), one angle gives all eight. Give a reason for each: "$f = 70^\circ$, corresponding to $b$". If the lines are not parallel, these rules do not hold.

Most angle problems take more than one step. For each step:

1. Know: an angle you already have.
2. Choose: a rule that links it to a new angle: corresponding or alternate angles are equal, co-interior angles add to $180^\circ$, vertically opposite angles are equal, angles on a straight line add to $180^\circ$.
3. Say why: write the rule as the reason.

$b = 70^\circ$: $f = 70^\circ$ (corresponding to $b$), then $h = 110^\circ$ ($f$ and $h$ are on a straight line). If two angles only look related, find one in between that a rule links to both.

The three angles of any triangle add to $180°$.

When you change a triangle's shape, one angle grows and the others shrink by exactly as much. Making it bigger without changing its shape does not change the angles at all.

Why? Tear off the three corners and put their points together: they fill a straight line, and angles on a straight line add to $180°$.

To find the angle sum of a polygon, split it into triangles from one corner: draw lines from that corner to every corner that is not next to it.

A shape with $n$ sides splits into $n - 2$ triangles, and their angles exactly fill the polygon's corners. So the angle sum is $(n - 2) \times 180°$.

A quadrilateral: $2 \times 180° = 360°$. A pentagon: $3 \times 180° = 540°$. A hexagon: $4 \times 180° = 720°$.

To find an unknown angle in a shape, choose the angle sum for the shape: $180^\circ$ for a triangle, $(n - 2) \times 180^\circ$ for $n$ sides ($360^\circ$ for a quadrilateral, $540^\circ$ for a pentagon). Add the angles you know and subtract from the sum.

Combine it with other facts, one step at a time, each with a reason. Extend a side of a triangle and the outside angle $x$ makes a straight line with $c$: $x = 180^\circ - c$. Since $a + b = 180^\circ - c$ too, $x = a + b$.

Name a triangle by its sides: scalene (none equal), isosceles (two equal), equilateral (all three equal). And by its biggest angle: acute (all under 90°), right-angled (one is 90°), obtuse (one over 90°).

Equal sides sit opposite equal angles. So an equilateral triangle has three 60° angles, and can never be right-angled or obtuse.

A quadrilateral's name comes from its properties: a trapezium has at least one pair of parallel sides; a parallelogram two; a rhombus four equal sides; a rectangle four right angles; a square four equal sides and four right angles; a kite two pairs of equal sides next to each other.

A shape with all of another name's properties belongs to that family too: every square is a rectangle and a rhombus, every rhombus and rectangle is a parallelogram, and every parallelogram is a trapezium.

A polygon is named by its number of sides: pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10.

It is regular when all its sides and all its angles are equal (a rhombus has equal sides but is not regular). It is concave when a corner points in (an angle over 180°), and convex when every corner points out.

A net is a solid unfolded flat. A cube's net is six squares that fold to cover all six faces once; six squares alone is not enough, so test by folding.

A triangular prism's net is two triangles (the ends) and three rectangles (one for each side of the triangle). Its area is the surface area.

A view is what you see from one side. The top view (plan) looks down; the front and side views (elevations) look across. They show the tallest pile in each column, and hide cubes behind or beneath.

A top view with a number on each pile shows every cube. A net shows every face at its true size, but not what the solid looks like when built.

A third axis, z, points up from the x-y plane. A point in space is (x, y, z): along x, along y, then up. The order is always x, y, z; points with z = 0 are on the floor.

Real systems work the same way: east, north and up for a drone; across, back and floor in a building; latitude, longitude and altitude for a plane.

A translation slides every point of a shape the same distance in the same direction. The object becomes the image, and A becomes A′.

Right and up add; left and down take away. A translation of 3 right and 2 down changes $(x, y)$ to $(x + 3, y - 2)$.

A reflection flips a shape over a mirror line; each point and its image are the same distance from the mirror, on opposite sides.

In the x-axis, $(x, y) \to (x, -y)$. In the y-axis, $(x, y) \to (-x, y)$.

A rotation turns a shape about a centre by an angle in a direction (clockwise or anticlockwise). The shape keeps its size.

About the origin, anticlockwise: 90° changes $(x, y)$ to $(-y, x)$; 180° changes it to $(-x, -y)$; 270° changes it to $(y, -x)$.

Congruent triangles are the same shape and size. Each test fixes a triangle: SSS, SAS (the angle between the sides), ASA or AAS, and RHS.

SSA (an angle not between the two sides) is not a test: it can give two different triangles. Equal angles alone give the same shape, not always the same size.

Similar shapes are the same shape, maybe a different size. Every length is multiplied by the same scale factor k = copy ÷ original, and the angles stay the same. Congruent is similar with k = 1.

Triangles are similar by AA, SSS (sides in one ratio) or SAS (two sides in one ratio, the angle between them equal). For other shapes, check both the angles and the ratios of the sides.

A proof gives a reason for every step and works for every shape of the kind; one measured drawing is only an example.

To prove a quadrilateral property: find two triangles (often either side of a diagonal), give the equal parts with reasons, name the congruence test, then use matching parts of congruent triangles. This proves, for example, that a rectangle's diagonals are equal.

Use the properties, with a reason for each step. A parallelogram's opposite angles are equal and its neighbouring (co-interior) angles add to 180°. A kite has one pair of equal opposite angles. A rectangle's diagonals are equal and cut each other in half; a rhombus's cut each other in half at 90°.

Every quadrilateral's angles add to 360°: use it to check.

An algorithm is a precise series of steps and decisions. A classification algorithm asks yes/no questions about attributes and ends in one name for each shape.

Put the most specific question first, make sure every branch ends in a name, and test it on one shape of every class. A flowchart, a decision tree and numbered steps show the same algorithm.

An algorithm for two triangles asks about their equal parts in order: the congruence tests first (SSS, SAS, ASA or AAS, RHS), then similarity (AA, sides in the same ratio), then not enough to tell.

Test it on one pair of every kind, including SSA (which must not be called congruent). When you fix it, test every pair again.