KS3 Maths · Year 8 · Chapter 3: Polygons · Lesson 1

Properties of polygons

Year 8About 60 minutesNo calculator46 marks

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Date

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Quick questions to start.

A1
Needed today
Two angles on a straight line. One is $4 7^{\circ}$ . Work out the other.
[1 mark]
Answer
A2
Needed today
Two angles of a triangle are $5 8^{\circ}$ and $7 1^{\circ}$ . Work out the third.
[1 mark]
Answer
A3
Last lesson
Solve $\frac{x}{4} + 3 = 9$ .
[1 mark]
Answer
A4
From chapter 1
Increase £ $60$ by $15%$ .
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

The diagonals from $A$ split the hexagon into triangles. Work out the sum of the interior angles of a hexagon.

Hint Count the triangles. The angles of each triangle add up to

18 0^{\circ}

B1
(a) Work out the sum of the interior angles of a decagon (10 sides).
(b) Write a formula for the sum of the interior angles of a polygon with $n$ sides.
[3 marks]
Answer(a) $\dots \dots^{\circ}$ (b) $\dots\dots$

Example 2

Work out the size of angle $a$ in this pentagon.

Not drawn accurately

Hint First work out what the five angles of a pentagon add up to.

B2
Work out the size of angle $b$ in this hexagon.
[2 marks]
Not drawn accurately
Answer $b = \dots \dots^{\circ}$

Example 3

Each side of this pentagon is extended to show its exterior angles. Work out the exterior angle $e$ , then the interior angle at that vertex.

Not drawn accurately

Hint The exterior angles of any polygon add up to

36 0^{\circ}

. At each vertex, interior + exterior

= 18 0^{\circ}

B3
Each side of this hexagon is extended. Work out the exterior angle $g$ , then the interior angle at that vertex.
[3 marks]
Not drawn accurately
Answer $g = \dots \dots^{\circ}$ interior $\dots \dots^{\circ}$

Example 4

In pentagon $A B C D E$ , angle $A = 2 x^{\circ}$ , $B = (x + 40)^{\circ}$ , $C = (3 x - 20)^{\circ}$ , $D = (2 x + 15)^{\circ}$ and $E = (x + 55)^{\circ}$ . Work out the value of $x$ and the size of the largest angle.

Not drawn accurately

Hint Add the five expressions and set the total equal to the angle sum of a pentagon.

B4
In hexagon $A B C D E F$ , angle $A = (x + 30)^{\circ}$ , $B = 2 x^{\circ}$ , $C = (2 x - 10)^{\circ}$ , $D = (x + 50)^{\circ}$ , $E = (3 x - 40)^{\circ}$ and $F = (3 x - 30)^{\circ}$ . Work out the value of $x$ and the size of the smallest angle.
[3 marks]
Not drawn accurately
Answer $x = \dots\dots$ smallest $\dots \dots^{\circ}$

Practice

Year 8

C1
Could the interior angles of a polygon add up to $100 0^{\circ}$ ? Give a reason for your answer.
[2 marks]
C2
At one vertex of a polygon, the interior angle is $4$ times the exterior angle. Work out the size of the exterior angle.
[2 marks]
Answer

C3
Write down the letters of the shapes that are concave polygons.
[2 marks]
Answer
C4
This quadrilateral is concave. Work out the size of the reflex angle $r$ .
[2 marks]
Not drawn accurately
Answer $r = \dots \dots^{\circ}$
C5
The interior angles of a polygon add up to $198 0^{\circ}$ . How many sides does the polygon have?
[2 marks]
Answer $\dots\dots$ sides
C6
Two angles of this pentagon are equal. Work out the size of angle $y$ .
[2 marks]
Not drawn accurately
Answer $y = \dots \dots^{\circ}$

Exam-style questions

GCSE Foundation

D1
Tom says, "The interior angles of a hexagon add up to $72 0^{\circ}$ , so its exterior angles add up to $72 0^{\circ}$ too."
Explain why Tom is wrong.
[2 marks]

D2
$A B C D E$ is a pentagon. Angle $A$ and angle $B$ are right angles.
Not drawn accurately
(a)
Work out the value of $x$ . You must show your working.
[3 marks]
Answer $x = \dots\dots$
(b)
Work out the size of the exterior angle at $D$ .
[2 marks]
Answer
Total for D2: 5 marks
D3
The five interior angles of a pentagon go up by $1 4^{\circ}$ each time, from the smallest to the largest. Show that the largest angle is $13 6^{\circ}$ .
[4 marks]
D4
The sides of quadrilateral $A B C D$ are extended to show its exterior angles. The exterior angles at $A$ , $B$ , $C$ and $D$ are $x^{\circ}$ , $2 x^{\circ}$ , $(x + 30)^{\circ}$ and $(2 x - 30)^{\circ}$ . Work out the size of the smallest interior angle of $A B C D$ .
[4 marks]
Not drawn accurately
Answer

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
A diagonal joins two vertices of a polygon that are not next to each other. A pentagon has $5$ diagonals. A polygon has $65$ diagonals. How many sides does it have?
[4 marks]
Hint 1 · What to try
Work out how many diagonals leave ONE vertex of a polygon with $n$ sides.
Hint 2 · The first line
Each vertex has $n - 3$ diagonals, and each diagonal is counted twice (once from each end), so there are $\frac{n ( n - 3 )}{2}$ diagonals.
Hint 3 · The full method
$n (n - 3) = 130$ . Try values: $13 \times 10 = 130$ , so $n = 13$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the sum of the interior angles of any polygon.
I can find a missing interior or exterior angle.
I can write and solve an equation from the angles of a polygon.
I can tell a convex polygon from a concave one.

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