GCSE Maths Geometry and Measure

2D Shapes

Polygons

Types of Triangles

# Types of Triangles

Here we will learn about types of triangles, including their names, their properties and applying them to problems.

There are also types of triangles worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What is a triangle?

A triangle is a 2D shape with 3 straight sides and 3 vertices. You need to be able to classify triangles based on their properties, and find unknown angles and side lengths.

### What is a triangle? ## Types of triangles

There are 5 different types of triangle. Below is a table that describes the properties of each triangle.

Properties of triangles are very important within geometry. They are one of the most common shapes to recognise for angles in parallel lines, circle theorems, interior angles, trigonometry, Pythagoras’ theorem, and many more. You therefore must be familiar with their individual properties.

## Angles in a triangle

Triangles can contain acute angles, a right angle, or one obtuse angle.

A triangle cannot contain more than one right angle, obtuse angle, or a reflex angle. The only two types of triangles that can contain an obtuse angle are an isosceles triangle and a scalene triangle.

E.g.

Here are two examples of these obtuse triangles:

The sum of angles in a triangle is 180° .

This property helps to find angles in any polygon as a polygon can be triangulated (split into triangles) and therefore the interior angles can be calculated.

For more information, see the lessons on angles on a triangle, interior angles and exterior angles.

## How to classify a triangle

In order to classify a triangle:

1. Determine the size of the angles/side lengths within the triangle.
2. Recognise the other properties of the triangle.

### How to classify a triangle ## Types of triangles examples

### Example 1: triangle

Correctly classify the following polygon:

1. Determine the size of the angles/side lengths within the triangle.

Angles in a triangle total 180° , so the missing angle is equal to:

180 - (70+55)=55° .

2Recognise the other properties of the triangle.

As two angles are the same length, the missing side must be equal to 7cm . The triangle also has one line of symmetry (vertical) and no rotational symmetry.

The polygon is an isosceles triangle.

### Example 2: multiple triangles

Two congruent, isosceles triangles are joined to form another triangle.

Use the information in the diagram below to classify the triangle ABD :

As ACD is an isosceles, the angle CAD = 45° .

The angle ACD therefore equals 180 - (45 + 45) = 90° .

As the triangles are congruent, angle BCD = 90° and the angles CBA and CAB = 45° .

This means the angle BAD = 45 + 45 = 90° .

As the angles BCD and ACD are both equal to 90° , BCD is a straight line.

The triangle ABD contains the three angles 90°, 45° and 45° so the triangle ABD is a right-angled isosceles triangle.

### Example 3: exterior angles

The exterior angles of a triangle are equal to 110°, 120° and 130° .

Classify the triangle.

As the sum of the interior and exterior angle at a vertex totals 180° , the interior angles are equal to:

180 - 110 = 70°

180 - 120 = 60°

180 - 130 = 50°

As all three angles in the triangle are different and so all sides are different lengths, the triangle is a scalene triangle.

### Example 4: algebraic

Three interior angles of a triangle are given as 2x+11, 3x+18 and x+31 .

Use this to determine the classification of the triangle.

As the sum of angles in a triangle total 180° ,

2x+11+3x+18+x+31=180

6x+60=180

6x=120

x=20

As x=20

2x+11=220+11=51°

3x+18=320+18=78°

x+31=20+31=51°

As two angles are equal, the triangle is an isosceles triangle.

### Example 5: description

Three people stood at different points in a field, each with a compass. The bearing of person B from person A is 120° . The bearing of person C from person B is 240° . Person C is standing directly south of person A .

If each person stood at the vertex of a triangle, what classification of triangle would they produce?

Drawing a sketch of the information above we have:

We can therefore find the three interior angles of the triangle ABC .

The interior angle at A = 180 - 120 = 60° (angles on a straight line total 180°).

The back bearing of A from B is equal to 180 - 120 = 60° (co-interior angles in parallel lines total 180° ).

This means the interior angle at B = 360 - (240 + 60) = 60° (angles at a point total 360° ).

The final angle at C can be calculated using the sum of angles in a triangle: 180 - (60 + 60) = 60° .

As all three angles are equal to 60° , the triangle is an equilateral triangle.

### Example 6: inscribed in a circle

The triangle ABC is inscribed in a circle.

Classify the triangle OBC .

As the angle at the centre is twice the angle at the circumference, the angle BOC = 46×2 = 92° .

OB and OC are radii of the circle and so triangle OBC is an isosceles triangle.

### Common misconceptions

• Angles in polygons

Make sure you know your angle properties. Getting these confused causes quite a few misconceptions.

• Angles in a triangle total 180°.
• Angles in a quadrilateral total 360°.
• Angles on a straight line total 180°.

• Incorrect assumptions for triangles

Assuming a triangle is isosceles or equilateral can have an impact on the size of angles within the rest of the polygon, so make sure you can explain why you have chosen a specific type of triangle.

As well as this, stating that the polygon is a triangle is not enough information for a classification. You must state whether it is isosceles/ equilateral etc.

### Practice types of triangles questions

1. Classify the following triangle. Isosceles triangle Scalene triangle Right angle isosceles triangle Right angle triangle 180 (52 + 38) = 90^{\circ} (angles in a triangle total 180^{\circ} )

2. Use the information in the diagram to determine the type of triangle of ABD . Right angle isosceles triangle Isosceles trapezium Scalene triangle Equilateral triangle BCD = 180(110+40) = 30^{\circ} (angles in a triangle total 180^{\circ} )

ADB = 180(75 + 30 + 40) = 35^{\circ} (angles in a triangle total 180^{\circ} )

ABD = 180110 = 70^{\circ} (angles on a straight line total 180^{\circ} )

All three angles in triangle ABD are different. This is a scalene triangle.

3. The exterior angles of a triangle are equal to 120^{\circ} each. Classify the triangle.

Equilateral triangle Isosceles triangle Scalene triangle Right angle isosceles triangle As all the exterior angles are equal, the interior angles are also equal at

180 120 = 60^{\circ} .

4. Three interior angles of a triangle are given as 7x+21, 3x+12, and 2x+15 . Use this to determine the classification of the triangle.

Equilateral triangle Isosceles triangle Scalene triangle Right angle triangle As the sum of angles in a triangle total 180^{\circ} ,

7x+21+3x+12+2x+15=180\\ 12x+48=180\\ 12x=132\\ x=11

As x=11
7x+21=711+21=98^{\circ}\\ 3x+12=311+12=45^{\circ}\\ 2x+15=211+15=37^{\circ}

5. A plane travels on a bearing of 105^{\circ} from point A , then at a bearing of 140^{\circ} from point B . The plane lands at point C . The point C is at a bearing of 137^{\circ} from point A . Classify the triangle made between points A, B, and C .

Right angle isosceles triangle Scalene triangle Isosceles triangle Right angle triangle  Angle BAC = 137 – 105 = 32^{\circ}

The back bearing at B = 180 – 105 = 75^{\circ}

Angle ABC = 360 – (170 + 75) = 115^{\circ} (angles at a point total 360^{\circ} )

Angle BCA = 180 – (115+32)=33^{\circ} (angles in a triangle total 180^{\circ} )

6. Use circle theorems to classify the triangle ABD . Isosceles triangle Equilateral triangle Scalene triangle Right angle triangle Two tangents that meet the circumference of a circle are the same length, so

AB = AD . This means that ABD is an isosceles triangle.

### Types of triangles GCSE questions

1.

(a) Below is a grid with three coordinates plotted. What type of triangle is ABC ?

(b) Use the grid to find the coordinate of the centre of the polygon.

(3 marks)

(a)

Isosceles triangle

(1)

(b) (1)

(3,3)

(1)

2.

(a) The rectangle ABCD and the triangle ADE are joined together as shown below. The lines AB and CE are parallel. Show that triangle ADE is a right angle isosceles triangle. (b) The line AC is joined to create a new triangle ACE with angle BAC = 42^{\circ} . Calculate the size of angle EAC . Explain your working.

(7 marks)

(a)

As AB and CE are parallel, angle ADE = 90^{\circ}

(1)

Angle DAE = 180 – (90+45) = 45^{\circ}

(1)

Angle DEA = angle DAE so the triangle is isosceles and Angle ADE is a right angle

(1)

(b)

Angles BAC and ACD are alternate angles in parallel lines which are equal

(1)

Angle ACE = 42^{\circ}

(1)

Angle EAC = 180 – (45+42) = 93^{\circ}

(1)

Angles in a triangle total 180^{\circ}

(1)

3.

(a) An isosceles triangle ABC contains the angle 50^{\circ} . Work out the possible sizes for the other two angles.

(b) A different isosceles triangle PQR is inscribed in a circle. The points P and R are connected to the centre of the circle to form a further triangle POR . What type of triangle is POR ? Explain your answer. (6 marks)

(a)

(180 -50)\div2 = 65^{\circ}

(1)

65^{\circ} and 65^{\circ}

(1)

180 (50+50) = 80^{\circ}

(1)

50^{\circ} and 80^{\circ}

(1)

(b)

Isosceles triangle

(1)

OP and OR are radii and so are the same length

(1)

## Learning checklist

You have now learned how to:

• Plot specified points and draw sides to complete a given polygon. Distinguish between regular and irregular polygons based on reasoning about equal sides and angles
• Compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
• Describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric
• Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons

## The next lessons are

•  Regular polygons
•  Irregular polygons