Angle Properties of Lines and Polygons

Angles measure turns

An angle measures the turn between two rays meeting at a point. A complete turn is $360^\circ$, a half-turn is $180^\circ$ and a right angle is $90^\circ$. These turns explain two rules used throughout geometry: angles around a point add to $360^\circ$, and adjacent angles making a straight line add to $180^\circ$.

To find a missing angle, identify the complete turn or straight angle, then subtract the angles already known. For example, if four angles around a point are $50^\circ$, $155^\circ$, $97^\circ$ and $$, then

$$x=360^\circ-50^\circ-155^\circ-97^\circ=58^\circ.$$

Sometimes an unknown appears more than once. If three adjacent angles on a straight line are $x$, $54^\circ$ and $x$, their equation is $2x+54^\circ=180^\circ$. The two equal angles share the remaining , so each is .

Intersecting lines: opposite angles are equal

Two straight lines crossing produce four angles. Each angle has a partner directly opposite it: these are vertically opposite angles, and they are equal. ‘Vertically’ here does not mean that the diagram must be upright.

This equality follows from the straight-line rule. Each of two opposite angles adds to $180^\circ$ with the same neighbouring angle, so the opposite angles must have the same size. If one angle is $70^\circ$, its opposite is also $70^\circ$, while each neighbouring angle is .

Parallel lines: recognise the position of each angle

Parallel lines never meet and stay the same distance apart. A line crossing them is called a transversal. The angle relationships depend on where the angles sit, not on whether the diagram is horizontal, tilted or reflected.

Corresponding angles occupy matching positions at the two crossings: for example, above each parallel line and to the right of the transversal. They are equal. An ‘F’ shape can help you recognise them.

Alternate interior angles lie between the parallel lines, on opposite sides of the transversal. They are equal. Their connecting lines form a ‘Z’ shape.

Co-interior angles lie between the parallel lines, on the same side of the transversal. They add to $180^\circ$. An alternate-angle equality and the straight-line rule explain this: one co-interior angle equals an angle adjacent to the other.

For example, if a corresponding angle to a given $63^\circ$ angle is needed, it is also $63^\circ$. An angle adjacent to it on a straight line is then $180^\circ-63^\circ=117^\circ$. This is a typical two-step deduction: first transfer an angle using parallel lines, then subtract using a straight line.

The corresponding, alternate and co-interior rules require parallel lines. Vertically opposite angles and the straight-line rule do not.

Triangles: a total of 180°

The three interior angles of every triangle add to $180^\circ$:

$$a+b+c=180^\circ.$$

To see why, draw a line through the top vertex parallel to the base. Alternate angles place copies of the two base angles beside the top angle. Together they make a straight angle, so their total is $180^\circ$.

A triangle’s side properties can give extra angle information. An isosceles triangle has two equal sides, and the angles opposite those sides are equal. An equilateral triangle has three equal sides and three equal angles; each angle is $180^\circ\div3=60^\circ$. A scalene triangle has no equal sides or equal angles.

A right-angled triangle has one $90^\circ$ angle, so the other two angles add to $90^\circ$. A triangle can be both right-angled and isosceles; its two remaining angles are then $45^\circ$ each.

Extending one side of a triangle creates an exterior angle. It adds to $180^\circ$ with the adjacent interior angle. Since the three interior angles also total $180^\circ$, the exterior angle equals the sum of the two non-adjacent interior angles.

Quadrilaterals: combine the total with the shape’s properties

A diagonal divides a quadrilateral into two triangles. Its four interior angles therefore add to $2\times180^\circ=360^\circ$. This applies to irregular quadrilaterals as well as familiar named shapes.

Special quadrilaterals provide additional information:

The parallel-line rules explain the parallelogram’s angle properties. A rectangle and a rhombus are special parallelograms, while a square has the properties of both a rectangle and a rhombus.

An isosceles trapezium has a line of symmetry perpendicular to its parallel sides. Reflection in this line matches the two angles at each parallel side, explaining why those angles are equal.

In a kite, take care to identify the correct equal pair: not both pairs of opposite angles are necessarily equal. Its line of symmetry joins the two vertices where equal sides meet and bisects the angles at those vertices, dividing each into two equal parts.

For example, if the equal opposite angles are each $64^\circ$ and another angle is $158^\circ$, the remaining whole interior angle is

$$360^\circ-64^\circ-64^\circ-158^\circ=74^\circ.$$

The kite’s line of symmetry bisects that remaining angle, so each half is $37^\circ$. This distinguishes a whole vertex angle from a labelled part of it.

Polygons: build the angle total from triangles

A polygon is a closed shape with straight sides. For a convex polygon, where every interior angle is less than $180^\circ$, draw diagonals from one vertex to all non-adjacent vertices. An $n$-sided polygon splits into $n$ triangles, giving

$$\text{Sum of interior angles}=(n-2)\times180^\circ.$$

A pentagon splits into three triangles, so its interior angles total $540^\circ$. A hexagon splits into four, giving $720^\circ$. A decagon has ten sides and splits into eight triangles, giving $1440^\circ$. These are totals, not the sizes of individual angles.

In a regular polygon, all sides and all interior angles are equal. Only then can you divide the total by the number of sides to find each interior angle:

$$\text{Each interior angle}=\frac{(n-2)\times180^\circ}{n}.$$

Exterior angles: one complete turn

An exterior angle lies between a side and the extension of its neighbouring side. The adjacent interior and exterior angles make a straight angle:

$$\text{Interior angle}+\text{exterior angle}=180^\circ.$$

Imagine walking around a convex polygon, turning at every corner. After one circuit you face your original direction again: the exterior turns total $360^\circ$. Take one exterior angle at each vertex, turning consistently in the same direction.

For a regular polygon these turns are equal, so

$$\text{Each exterior angle}=\frac{360^\circ}{n}.$$

You can reverse this relationship to find the number of sides. For example, a regular polygon with interior angles of $135^\circ$ has exterior angles of $180^\circ-135^\circ=45^\circ$. Its number of sides is , so it is an octagon.

Link reasons together

Angle problems often need several rules rather than one formula. Identify the straight lines, parallel pairs, equal sides and any named shapes. Use these to find an intermediate angle, then carry that information into a triangle, quadrilateral or polygon total. Each step should explain why an angle is equal to another, or why a group of angles has a particular total.

Lines

• Around a point: $360^\circ$.
• On a straight line: $180^\circ$.
• Vertically opposite angles: equal.