Points, Lines and Angles

Lessons

Senior 1 · Mathematics

Points, Lines and Angles

5 lessons

Lesson 5 of 5

Angles in triangles and polygons

25 min

Why does a STOP sign have eight corners?

Road signs in Rwanda use shapes that drivers recognise at a glance: a triangle warns, a circle orders, and the STOP sign is the only sign with eight equal sides. Each shape is a polygon, and the angles at its corners obey exact rules. The most important rule, that the angles of any triangle add up to $180^\circ$ , can be proved with the parallel-line rules of Lesson 4, and from triangles it extends to every polygon.

Definition

Polygon: a closed plane figure made of straight line segments (its sides) that meet only at their endpoints (its vertices). A polygon with $n$ sides also has $n$ vertices and $n$ interior angles.

Interior angle: the angle inside the polygon between two sides that meet at a vertex.

Convex polygon: a polygon whose interior angles are all less than $180^\circ$ . All the polygons in this lesson are convex.

The angles of a triangle

At $A$ , the angles $x$ , $y$ and $z$ lie side by side on a straight line.

Important

Theorem: the interior angles of a triangle add up to $180^\circ$

x + y + z = 180^\circ

Proof. Through $A$ , draw the line parallel to $B C$ .

The angle between this line and $A B$ equals $x$ (alternate angles, since the line is parallel to $B C$ ).
The angle between this line and $A C$ equals $z$ (alternate angles).
At $A$ , the three angles $x$ , $y$ and $z$ lie side by side on the straight line, so $x + y + z = 180^\circ$ (angles on a straight line).

The proof never used the actual sizes of the angles, so it works for every triangle.

Definition

Exterior angle of a triangle: the angle between one side of the triangle and the extension of the side next to it.

Interior opposite angles: the two interior angles of the triangle that are not next to that exterior angle.

Example: in the figure below, $\angle ACD$ is an exterior angle, and its interior opposite angles are $\angle A$ and $\angle B$ .

The exterior angle $\angle ACD = 115^\circ = 45^\circ + 70^\circ$ , the two interior angles opposite it.

Important

Theorem: an exterior angle of a triangle equals the sum of the two interior opposite angles

\angle ACD = \angle A + \angle B

Why: $\angle ACD + \angle ACB = 180^\circ$ (angles on a straight line), and $\angle A + \angle B + \angle ACB = 180^\circ$ (angle sum of a triangle). Both sums equal $180^\circ$ and both contain $\angle ACB$ , so $\angle ACD = \angle A + \angle B$ .

Worked example 1: a triangle and its exterior angle

Find x and y.

Solution

x = 180^\circ - 48^\circ - 67^\circ = 65^\circ \quad \text{(angle sum of a triangle)}

y = 48^\circ + 65^\circ = 113^\circ \quad \text{(exterior angle of a triangle)}

Check: $y + 67^\circ = 113^\circ + 67^\circ = 180^\circ$ (angles on a straight line). ✓

From triangles to every polygon

Polygons are named by their number of sides. Pick one vertex and draw every diagonal from it: the polygon falls apart into triangles, and there are always two fewer triangles than sides.

From one vertex, an $n$ -sided polygon splits into $n - 2$ triangles.

Important

Theorem: the interior angles of a polygon with $n$ sides add up to

S = (n - 2) \times 180^\circ.

Why: the diagonals from one vertex split the polygon into $n - 2$ triangles, and the angles of these triangles together make up exactly the interior angles of the polygon. Each triangle contributes $180^\circ$ .

Sides $n$	Name	Angle sum $(n-2) \times 180^\circ$	Each angle if regular
3	triangle	$180^\circ$	$60^\circ$
4	quadrilateral	$360^\circ$	$90^\circ$
5	pentagon	$540^\circ$	$108^\circ$
6	hexagon	$720^\circ$	$120^\circ$
7	heptagon	$900^\circ$	about $128.6^\circ$
8	octagon	$1080^\circ$	$135^\circ$
9	nonagon	$1260^\circ$	$140^\circ$
10	decagon	$1440^\circ$	$144^\circ$

Definition

Regular polygon: a polygon whose sides are all equal and whose angles are all equal.

Example: a square is a regular quadrilateral; a rectangle that is not a square has equal angles but unequal sides, so it is not regular.

Definition

Exterior angle of a polygon: at a vertex, the angle between one side and the extension of the side before it. It is the angle you turn through at that corner when you walk round the polygon.

At every vertex, interior angle $+$ exterior angle $= 180^\circ$ (angles on a straight line).

Walk round the pentagon and turn at each corner: $65^\circ + 70^\circ + 75^\circ + 60^\circ + 90^\circ = 360^\circ$ , one full turn.

Important

Theorem: the exterior angles of any convex polygon add up to $360^\circ$

Why: walk once round the polygon. You turn through each exterior angle once and end up facing the way you started: one full turn. Or with algebra: the $n$ interior-plus-exterior pairs make $n \times 180^\circ$ , and the interior angles alone make $(n - 2) \times 180^\circ$ , which leaves $2 \times 180^\circ = 360^\circ$ for the exterior angles.

So in a regular polygon with $n$ sides,

\text{each exterior angle} = \frac{360^\circ}{n},

\text{each interior angle} = 180^\circ - \frac{360^\circ}{n} = \frac{(n-2) \times 180^\circ}{n}.

Worked example 2: a missing angle in a pentagon

A pentagon with one unknown angle.

Solution For a pentagon, $n = 5$ , so the angles add up to $(5 - 2) \times 180^\circ = 540^\circ$ .

x = 540^\circ - (100^\circ + 110^\circ + 120^\circ + 95^\circ) = 540^\circ - 425^\circ = 115^\circ.

Worked example 3: the STOP sign

A STOP sign is a regular octagon: 8 equal sides and 8 equal angles.

Find each interior angle and each exterior angle of a regular octagon.

Solution The interior angles add up to $(8 - 2) \times 180^\circ = 1080^\circ$ , and all eight are equal:

\text{interior angle} = \frac{1080^\circ}{8} = 135^\circ, \qquad \text{exterior angle} = \frac{360^\circ}{8} = 45^\circ.

Check: $135^\circ + 45^\circ = 180^\circ$ . ✓

Worked example 4: how many sides?

(a) Each exterior angle of a regular polygon is $24^\circ$ . How many sides does it have?
(b) Can a regular polygon have interior angles of $130^\circ$ ?

Solution

The exterior angles add up to $360^\circ$ , so $n = \frac{360^\circ}{24^\circ} = 15$ . It has 15 sides.
Each exterior angle would be $180^\circ - 130^\circ = 50^\circ$ , so $n = \frac{360^\circ}{50^\circ} = 7.2$ . A polygon cannot have 7.2 sides, so no regular polygon has interior angles of $130^\circ$ .

Warning

Common mistakes

Using $n \times 180^\circ$ instead of $(n - 2) \times 180^\circ$ . A quick test: for a triangle, $n = 3$ must give $180^\circ$ .
Dividing by $n$ for a polygon that is not regular. "Each angle $= \frac{(n-2) \times 180^\circ}{n}$ " is true only when all the angles are equal.
Marking the reflex angle outside a vertex as the exterior angle. The exterior angle is between a side and the extension of the next side, and it is less than $180^\circ$ in a convex polygon.

Tip

For the angle sum, remember "take away two, then times 180". For a regular polygon, find the exterior angle first ( $360^\circ \div n$ ); it is quicker, and the interior angle is $180^\circ$ minus it.

Khan Academy: Sum of interior angles of a polygon (9 min). Watch how Sal splits polygons into triangles from one vertex and finds the pattern (n − 2) × 180°.

✦

Practice

Give reasons. The last three questions need more than one step.

Two angles of a triangle are $35^\circ$ and $90^\circ$ . Find the third, and name the type of triangle by its largest angle.
Find the sum of the interior angles of (a) a hexagon (b) a decagon (c) a polygon with 12 sides.
Find each interior angle of a regular (a) pentagon (b) nonagon.
An exterior angle of a triangle is $128^\circ$ , and one of its interior opposite angles is $57^\circ$ . Find the other interior opposite angle.
The angles of a quadrilateral are $x$ , $2 x$ , $3 x$ and $4 x$ . Find $x$ and the largest angle.
Each interior angle of a regular polygon is $156^\circ$ . How many sides does it have?
Uwase says a regular polygon can have exterior angles of $25^\circ$ . Is she right? Explain.

Show answersHide answers

$180^\circ - 35^\circ - 90^\circ = 55^\circ$ ; a right-angled triangle.
(a) $4 \times 180^\circ = 720^\circ$ (b) $8 \times 180^\circ = 1440^\circ$ (c) $10 \times 180^\circ = 1800^\circ$
(a) $\frac{540^\circ}{5} = 108^\circ$ (b) $\frac{1260^\circ}{9} = 140^\circ$ , or $180^\circ - \frac{360^\circ}{9} = 140^\circ$
$128^\circ - 57^\circ = 71^\circ$ (exterior angle of a triangle)
$10x = 360^\circ$ , so $x = 36^\circ$ ; the largest angle is $4x = 144^\circ$ .
Exterior angle $= 180^\circ - 156^\circ = 24^\circ$ , so $n = \frac{360^\circ}{24^\circ} = 15$ sides.
No: $\frac{360^\circ}{25^\circ} = 14.4$ , which is not a whole number of sides.

1Why does a STOP sign have eight corners?

2The angles of a triangle

3From triangles to every polygon

Why does a STOP sign have eight corners?

The angles of a triangle

From triangles to every polygon