Angles
Revision Notes
Angle Facts






Acute angles – less than 90o.
Right angle – equal to 90o.
Obtuse angles – greater than 90o but less than 180o.
Reflex angles – greater than 180o but less than 360o.
The angles on a straight line equal 180o.
The angles around a point or in a full turn equal 360o.
Angles in Parallel Lines
Vertically opposite angles –
Vertically opposite angles are equal.
Corresponding angles –
Corresponding angles are equal.
Alternate angles –
Alternate angles are equal.
Interior angles –
Interior angles add up to 180o.
e.g. Find the missing angles giving reasons for your answers.
a = 45o because it is on a straight line with 135o
b = 45o because it is alternate to angle a and interior to 135 o
c = 135o because it is corresponding with 135o and on a straight line with angle b
Interior and exterior angles in Polygons
To find what the interior angles of a polygon add up to, use the following formula -
𝐀𝐧𝐠𝐥𝐞 𝐬𝐮𝐦 𝐨𝐟 𝐚 𝐏𝐨𝐥𝐲𝐠𝐨𝐧 = (𝑵𝒐 𝒐𝒇 𝒔𝒊𝒅𝒆𝒔 − 𝟐) × 𝟏𝟖𝟎
e.g.
The angle sum of a Hexagon(6 sided shape) would be –
𝐀𝐧𝐠𝐥𝐞 𝐬𝐮𝐦 𝐨𝐟 𝐚 𝐏𝐨𝐥𝐲𝐠𝐨𝐧 = (𝟔 − 𝟐) × 𝟏𝟖𝟎
= 𝟒 × 𝟏𝟖𝟎
= 𝟕𝟐𝟎𝒐
You can use the angle sum of a polygon to find unknown angles in that polygon.
The exterior angle of a polygon and its corresponding interior angle always add up to 180°
(because they make a straight line).
e.g. Find the size of the interior and exterior angles in this regular polygon.
Interior angle – Angle sum = (5 – 2) X 180 = 540o
Each interior angle therefore =
540
5
= 108𝑜
Exterior angle = 180o – 108o = 72o
For any polygon, regardless of the number of sides it has,
the sum of its exterior angles is always 360°.
Calculating the number of sides in a regular polygon, given the interior angle
To calculate the number of sides in a regular polygon you need to use the following formula-
𝟑𝟔𝟎
𝑵 𝒐𝒇 𝒔𝒊𝒅𝒆𝒔 =
𝒔𝒊𝒛𝒆 𝒐𝒇 𝒆𝒙𝒕𝒆𝒓𝒊𝒐𝒓 𝒂𝒏𝒈𝒍𝒆
𝒐
e.g.
The interior angles of a regular polygon are each 120°. Calculate the number of sides.
First you need to work out the exterior angles of this polygon.
Remember that the sum of an interior angle and the corresponding exterior angle is 180 o.
Therefore, for this example the exterior angle will be 180o – 120o = 60o.
We know that the exterior angles of any polygon add up to 360o.
𝑵𝒐 𝒐𝒇 𝒔𝒊𝒅𝒆𝒔 =
𝟑𝟔𝟎
= 𝟔 𝒔𝒊𝒅𝒆𝒔
𝟔𝟎
Exam Questions
Q1.
ABC, PQR and AQD are straight lines.
ABC is parallel to PQR.
Angle BAQ = 35°
Angle BQA = 90°
Work out the size of the angle marked x.
Give reasons for each stage of your working.
x=......................°
(Total for Question is 4 marks)
Q2.
The diagram shows a square and 4 regular pentagons.
Work out the size of the angle marked x.
.......................................................................................................................................
(Total for Question is 3 marks)
Q3.*
ABC is parallel to DEF.
EBP is a straight line.
AB = EB.
Angle PBC = 40°.
Angle AED = x°.
Work out the value of x.
Give a reason for each stage of your working.
(Total for Question is 5 marks)
Q4.
The interior angle of a regular polygon is 160°.
Diagram NOT accurately drawn
(i) Write down the size of an exterior angle of the polygon.
......................°
(ii) Work out the number of sides of the polygon.
......................................................................................................................................
(Total for Question is 3 marks)
Q5.*
CDEF is a straight line.
AB is parallel to CF.
DE = AE.
Work out the size of the angle marked x.
You must give reasons for your answer.
(Total for Question is 4 marks)
Mock exam question