Angles

Naming angles and giving reasons

An angle is the amount of turn between two lines that meet at a point, the vertex. Angles are measured in degrees: a full turn is ${360}^{°}$.

• An acute angle is less than ${90}^{°}$; a right angle is exactly ${90}^{°}$; an obtuse angle is between ${90}^{°}$ and ${180}^{°}$; a reflex angle is between ${180}^{°}$ and ${360}^{°}$.

Three-letter notation. Angle $ABC$ (also written $\angle ABC$) is the angle at $B$, between the lines $BA$ and $BC$. The middle letter is always the vertex. Angle $ABC$ and angle $CBA$ are the same angle. When three or more lines meet at a point, one letter is not enough to say which angle you mean, so exam questions use three letters, and so should you.

Reasons. Many questions say "Give a reason for your answer" or "Give a geometrical reason". The mark is for the correct geometrical property, in the correct words. Learn these wordings:

• angles on a straight line add up to ${180}^{°}$
• angles at a point add up to ${360}^{°}$
• vertically opposite angles are equal
• angles in a triangle add up to ${180}^{°}$
• angles in a quadrilateral add up to ${360}^{°}$
• corresponding angles are equal; alternate angles are equal; co-interior angles add up to ${180}^{°}$

Informal names such as "Z angles" or "F angles" do not earn the reason mark. Neither does a calculation on its own: $180-128=52$ shows the working, but the reason is the fact that made you subtract from 180.

Angles at a point and on a straight line

Angles at a point add up to ${360}^{°}$, because together they make one full turn.

Angles on a straight line add up to ${180}^{°}$, because a straight line is half a turn. Two angles that add up to ${180}^{°}$ are called supplementary.

Vertically opposite angles are equal. When two straight lines cross, they make two pairs of opposite angles. Here is why each pair is equal. Call the four angles $a$, $b$, $c$ and $d$ going round. Then $a+b=180$ (straight line) and $b+c=180$ (straight line). Both equal 180, so $a+b=b+c$, and therefore $a=c$. The same argument gives $b=d$.

Notice what this means for the crossing: once you know one of the four angles, you know all of them. If one angle is ${38}^{°}$, its opposite angle is ${38}^{°}$ and the other two are each $180-38={142}^{°}$.

Angles in triangles and quadrilaterals

Angles in a triangle add up to ${180}^{°}$. You can see why in the parallel lines section below: a line through one vertex parallel to the opposite side moves all three angles onto one straight line.

Angles in a quadrilateral add up to ${360}^{°}$. A diagonal splits any quadrilateral into two triangles, and the angles of the two triangles together make up the angles of the quadrilateral: $2\times 180=360$.

Some special triangles have extra facts:

• In an isosceles triangle, the two angles opposite the equal sides (the base angles) are equal. Equal sides are shown on a diagram by matching tick marks.
• In an equilateral triangle every angle is ${60}^{°}$.
• The exterior angle of a triangle (between one side and the next side extended) equals the sum of the two interior angles opposite it. It comes from the two facts above: the exterior angle and the interior angle next to it add up to ${180}^{°}$, and so do the three interior angles.

Using algebra. When angles are given as expressions, write an equation from the angle fact, then solve it. For example, if the angles of a triangle are $x$, $2x+12$ and $3x-6$, then $x+2x+12+3x-6=180$, so $6x+6=180$ and $x=29$. Always check by putting your value back in: the angles $29$, $70$ and $81$ add up to $180$.

Angles and parallel lines

Parallel lines never meet: they stay the same distance apart. On a diagram they are shown with matching arrows. A line that crosses two parallel lines is called a transversal, and it makes eight angles, four at each crossing. Because the two lines are parallel, the transversal crosses each of them at the same angle, so the pattern of angles at the top crossing is a copy of the pattern at the bottom crossing. That is where the three facts come from.

Corresponding angles are equal. They are in matching positions at the two crossings: for example, both above their line and to the right of the transversal. They make an F shape.

Alternate angles are equal. They are between the parallel lines, on opposite sides of the transversal. They make a Z shape. Each one is vertically opposite the angle that corresponds to the other, which is why they are equal.

Co-interior angles add up to ${180}^{°}$ (they are supplementary). They are between the parallel lines, on the same side of the transversal, making a C or U shape. One of them is on a straight line with the angle corresponding to the other, which is why they add up to ${180}^{°}$.

The shapes F, Z and C help you spot the pairs, but in an answer you must write the proper names: corresponding, alternate, co-interior.

Why the angles of a triangle add up to ${180}^{°}$. Draw a line through one vertex, parallel to the opposite side. The two base angles reappear at the top as alternate angles, and the three angles at the top sit on a straight line, so together they make ${180}^{°}$.

Polygons: interior and exterior angles

A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.

At each vertex there is an interior angle (inside the shape) and an exterior angle (between one side and the next side extended). They sit together on a straight line, so at every vertex:

$$\text{interior angle}+\text{exterior angle}={180}^{°}$$

The exterior angles of a polygon add up to ${360}^{°}$. Imagine walking all the way round the edge. At each corner you turn through the exterior angle, and by the time you are back where you started, facing the same way, you have turned through one full turn.

The interior angles of a polygon with $n$ sides add up to $(n-2)\times {180}^{°}$. From one vertex, draw the diagonals to every other vertex. They split the polygon into $n-2$ triangles, each with an angle sum of ${180}^{°}$. A pentagon makes 3 triangles, so its angles add up to ${540}^{°}$; a hexagon makes 4, so ${720}^{°}$.

For a regular polygon with $n$ sides, all the angles are equal, so:

• each exterior angle $=360÷n$;
• each interior angle $=180-360÷n$, which is the same as $(n-2)\times 180÷n$.

Working backwards. To find the number of sides from an angle, always go through the exterior angle: $n=360÷\text{exterior angle}$. If you are given the interior angle, subtract it from 180 first. Because $n$ must be a whole number, the exterior angle of a regular polygon must divide exactly into 360. That is how you show that an angle is impossible: an interior angle of ${130}^{°}$ would need $360÷50=7.2$ sides.

Fitting shapes at a point. When polygons fit together round a point with no gaps, their angles at that point add up to ${360}^{°}$ (angles at a point). This is how tiling questions work.