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Types of angles Angles in a triangle Angle rulesThis topic is relevant for:

Here we will learn about **vertically opposite angles **including how to find missing angles which are vertically opposite each other at the same vertex.

There are also angles in polygons worksheets based on Edexcel, AQA and OCR GCSE exam style questions, along with further guidance on where to go next if you’re still stuck.

**Vertically opposite angles **are angles that are opposite one another at a specific **vertex** and are created by **two straight intersecting lines**.

Vertically opposite angles are **equal **to each other.

These are sometimes called vertical angles.

Here the two angles labelled **vertically opposite**’ at the same vertex. This also applies to the angles labelled

You can try out the above rule by drawing two crossing lines and measuring the angles opposite to one another.

You will also notice that angle

*Note because the sum of angles ‘a’ and ‘b’ are 180º we can call them supplementary angles.*

Before we start looking at specific examples it is important we are familiar with some **key words**, **terminology **and **symbols required **for this topic.

**Angle:**defined as the amount of turn around a common vertex.

**Vertex**: the point created by two line segments meeting (*plural is vertices)*

**How to label an angle**:

We normally label angles in two main ways:

1By giving the angle a ‘name’ which is normally a lower case letter such as

2By referring to the angle as the three letters that define the angle. The middle letter refers to the vertex at which the angle is e.g. see the diagram for the angle we call ABC:

**Angles on a straight line equal****180º****:**

Angles on one part of a straight line **always add up to **

However see the next diagram for an example of where **not **equal

*Note – you can try out the above rule by drawing out the above diagrams and measuring the angles using a protractor.*

**Angles around a point equal****360º****:**

Angles around a point will **always equal ****. **See the diagram for an example where angles

**Supplementary and Complementary Angles**:

Two angles are** supplementary** when they **add up to 180º, **they do not have to be next to each other (see below on ).

Two angles are **complementary** when they **add up to 90º, **they do not have to be next to each other (see the diagram)

In order to solve problems involving angles you should follow these steps:

**Identify which angles are vertically opposite to one another.**

Write this down e.g.a = b .**Clearly identity which of the unknown angles the question is asking you to find the value of.****Solve the problem and give reasons where applicable.****Clearly state the answer using angle terminology.**

Get your free vertically opposite angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free vertically opposite angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEFind the value of angle

**Identify which angles are vertically opposite to one another**.

The angle labelled

**2 Clearly identify which of the unknown angles the question is asking you to find the value of.**

The angle labelled

**3 Solve the problem and give reasons where applicable**.

**4 Clearly state the answer using angle terminology.**

Find the values of angles

**Identify which angles are vertically opposite to one another.**

The angle labelled

**Clearly identity which of the unknown angles the question is asking you to find the value of.**

The angles labelled

*You will notice that x and y are not vertically opposite one another*.

**Solve the problem and give reasons where applicable.**

\begin{aligned}
x+y &= 180 \hspace{3cm} \text{because they are on a straight line at the same vertex}\\\\
93+y &= 180 \hspace{3cm} \text{subtract 93 from each side} \\\\
y&=87
\end{aligned}

Clearly state the answer using angle terminology

Find the values of the angles labelled

**Identify which angles are vertically opposite to one another.**

The angle labelled

The angle labelled

**Clearly identity which of the unknown angles the question is asking you to find the value of.**

The angles labelled

**Solve the problem and give reasons where applicable.**

\begin{aligned}
a+b &= 180 \hspace{2cm} & \text{because they are on a straight line at the same vertex}\\\\
22+b &= 180 & \text{subtract 22 from each side} \\\\
b&=158
\end{aligned}

**Clearly state the answer using angle terminology.**

Using vertically opposite angles find the value of

**Identify which angles are vertically opposite to one another**.

The angle labelled

The angle labelled

**Clearly identity which of the unknown angles the question is asking you to find the value of.**

We are not being asked to find an angle we are being asked to find the value of

**Solve the problem and give reasons where applicable.**

The four angles total

\begin{aligned}
x+10+x+120+x+10+x+120 &= 360 \hspace{.5cm} & \text{simplify the equation}\\\\
4x+260 &= 360 & \text{subtract 260 from each side} \\\\
4x&=100 & \text{divide each side by 4} \\\\
x&=25
\end{aligned}

**Clearly state the answer using angle terminology.**

In the diagram below

Find the value of angle

**Identify which angles are vertically opposite to one another.**

The angle of size

**Clearly identity which of the unknown angles the question is asking you to find the value of.**

Find angle * at the top vertex of the triangle*.

**Solve the problem and give reasons where applicable.**

Angle

\begin{aligned}
B A C+80+80&=180 \\\\
BAC+160&=180 \\\\
BAC&=20
\end{aligned}

**Clearly state the answer using angle terminology.**

Angle

Two angles with values of

**Prove the two angles are both 50º**.

**Identify which angles are vertically opposite to one another.**

The angles labelled

**Clearly identity which of the unknown angles the question is asking you to find the value of.**

You are being asked to prove the size of each angle is

**Solve the problem and give reasons where applicable.**

\begin{aligned}
x+30&=4x-30 \hspace{.25cm} & \text{because the angles are vertically opposite to one another}\\\\
x+60&=4x & \text{add 30 to both sides} \\\\
60 &=3x & \text{subtract x from each side} \\\\
20 &=x & \text{divide each side by 3}
\end{aligned}

Therefore the size of the two angles can be found by substitution

Angle 1:

\begin{aligned}
x+30&=20+30\\\\
x&=50^{\circ}
\end{aligned}

Angle 2:

\begin{aligned}
4x-30&=4(20)-30\\\\
x&=50^{\circ}
\end{aligned}

**Clearly state the answer using angle terminology.**

Therefore both angles are

**Incorrectly labelling angles which are vertically opposite one another**

**Misuse of the ‘straight line’ rule where angles do not share a vertex**

**Finding the incorrect angle due to misunderstanding the terminology**

1. Find the value of the angle labelled x :

x=113

x=23

x=67

x=22

Angle x is vertically opposite the given angle of 67^{\circ} so it is the same.

2. Find the value of the angle labelled x :

x=146

x=56

x=34

x=214

Angle x is vertically opposite the given angle of 146^{\circ} so it is the same.

3. Find the value of the angle labelled x :

x=72

x=86

x=82

x=98

Angle x is vertically opposite the given angle of 98^{\circ} so it is the same.

4. Find the value of the angle labelled x and y :

x=68, y=68

x=112, y=112

x=112, y=68

x=68, y=112

Angle x is vertically opposite the given angle of 112^{\circ} so it is the same.

Angle x and angle y lie on a straight line so they must add up to 180 .

5. Two angles with values of 2x and 50^{\circ} are vertically opposite one another. Find the value of x .

x=50

x=25

x=100

x=75

Angle 2x is vertically opposite the given angle of 50^{\circ} so it is the same.

To solve for x , we divide 50 by 2 .

6. Two angles with values of 6x+10 and 10x-70 are vertically opposite one another. Find the value of x

x=20

x=40

x=70

x=10

Angle 6x + 10 is vertically opposite angle of 10x − 70 so they are the same.

Solving the equation, 6x + 10 = 10x − 70 , leads to the correct value for x .

1. Find the size of angles a and b .

**(2 marks)**

Show answer

a = 142^{\circ} (because vertically opposite angles are equal)

**(1)**

b: 180 − 142 = 38^{\circ} (because angles on a straight line add to 180)

**(1)**

a = 142^{\circ}, b = 38^{\circ}

2.

(a) Write an equation involving x.

(b) Use your equation to find the size of the angles.

**(4 marks)**

Show answer

(a)

2x + 14 = 3x − 5

**(1)**

(b)

14 = x − 5

**(1)**

x = 19

**(1)**

Angles:

2 \times 19 + 14 = 52

= 52^{\circ}

**(1)**

3. Prove that triangle ABC is a right angle triangle

**(3 marks)**

Show answer

Angle ACB = 24^{\circ} since they are vertically opposite

**(1)**

66 + 24 = 90

**(1)**

180 − 90 = 90 * *(angles in a triangle add up to 180 ),

so angle BAC is 90^{\circ} and this is a right angle triangle.

**(1)**

You have now learned how to:

- Use conventional terms and notation for angles
- Apply the properties of vertically opposite angles
- Apply angle facts and properties to solve problems

- Angles in parallel lines
- Corresponding angles
- Alternate angles
- Co-interior angles
- Angles in polygons
- Interior angles of a polygon
- Exterior angles of a polygon
- Trigonometry

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