Angles Point - Math Steps, Examples & Questions

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Math resources Geometry Angles

Angles point

Here you will learn about angles point, or angles around a point, including the sum of angles around a point, how to find missing angles, and using these angle facts to generate equations and solve problems.

Students will first learn about angles point as a part of geometry in $7$ th grade.

What are angles point?

Angles point describes the sum of angles that can be arranged together so that they form a full turn. Angles around a point add to $360^{\circ}.$

Let us look at this visually. Look at the five angles below:

Rearranging these angles so that they meet at one point or vertex, you get:

The sum of the angle measures is $360^{\circ}$ and they have made a full turn, and so the angles around a point add to equal $360^{\circ}.$

Angles around a point are applied to many problem solving style questions including properties of 2D shapes and 3D shapes including right angles, exterior angles, bearings, angles in parallel lines, pie charts, vertically opposite angles, and more.

What are angles point?

Common Core State Standards

How does this relate to $7$ th grade math?

Grade 7: Geometry (7.6.B.5)
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

[FREE] Angles Worksheet (Grade 4)

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support

DOWNLOAD FREE

[FREE] Angles Worksheet (Grade 4)

Use this quiz to check your grade 4 students’ understanding of angles. 10+ questions with answers covering a range of 4th grade angles topics to identify areas of strength and support

DOWNLOAD FREE

How to find missing angles around a point

In order to find missing angles around a point:

Add all known angles.
Subtract the angle sum from $\bf{360}^{\circ}.$
Form and solve the equation.

Angles point examples

Example 1: reflex angle

The lines $A$ and $B$ meet at the point $O.$ Calculate the missing angle $a.$

Add all known angles.

You only know one angle, measuring $128^{\circ}.$

2Subtract the angle sum from $\bf{360}^{\circ}.$

\begin{aligned}& 360-128=232^{\circ} \\\\ & a=232^{\circ} \end{aligned}

3Form and solve the equation.

This step is not needed for this example.

Example 2: multiple angles around a point

Calculate the size of angle $x.$

Add all known angles.

151+64+90=305^{\circ}

Subtract the angle sum from $\bf{360}^{\circ}.$

\begin{aligned}& 360-305=55^{\circ} \\\\ & x=55^{\circ} \end{aligned}

Form and solve the equation.

This step is not needed for this example.

Example 3: bearings

A ship is facing east. How far does the ship need to turn in the clockwise direction to face northwest?

Add all known angles.

The angle between northwest and north is a turn of $45^{\circ},$ since northwest is halfway between north and west, a $90^{\circ}$ angle.

$45+90=135^{\circ}$

Subtract the angle sum from $\bf{360}^{\circ}.$

\begin{aligned}& 360-135=225^{\circ} \\\\ & x=225^{\circ} \end{aligned}

Form and solve the equation.

This step is not needed for this example.

Example 4: reflex angle inside a polygon

$ABCD$ is an arrowhead. Calculate the missing angle $c.$

Add all known angles.

The sum of angles in a quadrilateral is equal to $360^{\circ}.$ Use this fact first to determine the reflex angle of $BCD.$

$\begin{aligned}& 16+27+27=70^{\circ} \\\\ & 360-70^{\circ}=290^{\circ} \end{aligned}$

The reflex angle at $C=290^{\circ}.$

Subtract the angle sum from $\bf{360}^{\circ}.$

\begin{aligned}& 360-290=70^{\circ} \\\\ & C=70^{\circ} \end{aligned}

Form and solve the equation.

This step is not needed for this example.

Example 5: forming and solving equations

The point $O$ is surrounded by five angles shown in the diagram below. Calculate the value of $x.$

Add all known angles.

8 x+4 x+5 x+8+5 x-7+7 x+11=29 x+12

Subtract the angle sum from $\bf{360}^{\circ}.$

This step is not needed for this example.

Form and solve the equation.

Angles around a point add to $360^{\circ}$ , so we have the equation $29 x+12=360.$

As $x=12,$ substitute this into each angle to find their values:

$\begin{aligned}& 4 x=4 \times 12=48^{\circ} \\\\ & 5 x+8=5 \times 12+8=68^{\circ} \\\\ & 5 x-7=5 \times 12-7=53^{\circ} \\\\ & 7 x+11=7 \times 12+11=95^{\circ} \\\\ & 8 x=8 \times 12=96^{\circ} \end{aligned}$

Check the solution by adding up the angles:

$48+68+53+95+96=360^{\circ}$

Example 6: angles in parallel lines

Calculate the exact value of $y.$

Add all known angles.

Using angle facts in parallel lines, you can state other angles in the diagram shown below:

This means there is a point surrounded by angles that you can add together:

$\begin{aligned}& 3 x-12+y+2 x+21+y=5 x+2 y+9 \\\\ & \text { So } 5 x+2 y+9=360 \\\\ & \text { or } 5 x+2 y=351 \end{aligned}$

Subtract the angle sum from $\bf{360}^{\circ}.$

This step is not needed for this example.

Form and solve the equation.

The vertically opposite angles of $3 x-12$ and $2 x+21$ are equal so you can solve:

Substituting $x=33$ into our equation from earlier you get:

$\begin{aligned}& 5 \times 33+2 y=351 \\\\ & 2 y=186 \\\\ & y=93^{\circ} \end{aligned}$

Teaching tips for angles point

Begin by introducing the concept visually, using the example of an analog clock, where each hour mark represents $30$ degrees.

While worksheets have their place within the classroom, consider using interactive games or quizzes that keep students interested and engaged. There are lots of online resources where students have the ability to manipulate angles in real-time.

Easy mistakes to make

Confusing the sum of angles point and angles on a straight line
The angle sum is remembered incorrectly as $180^{\circ},$ rather than $360^{\circ}.$ The sum of angles around a point is one full turn, or $360^{\circ}.$

Incorrectly assuming the diagram is accurately drawn
The reflex angle in a diagram is incorrectly assumed to be double the other angle around a point.
For example,

the reflex angle is incorrectly calculated as $128 \times 2=256^{\circ}.$

Using a protractor
When you are asked to calculate a missing angle, a common error is to use a protractor to measure the angle. When using angle rules to determine angles, diagrams are deliberately not drawn to scale, unless the angle is $90^{\circ}$ or $180^{\circ}.$ You would not use a protractor for this style of question.

Relate angles lessons

Practice angles point questions

256^{\circ}

76^{\circ}

208^{\circ}

104^{\circ}

Add up the measures of the given angles, then subtract from $360^{\circ}.$

360-104=256^{\circ}

24^{\circ}

112^{\circ}

22^{\circ}

12^{\circ}

Add up the measures of the given angles, then subtract from $360^{\circ}.$

182+90+66=338^{\circ}

360-338=22^{\circ}

45^{\circ}

225^{\circ}

270^{\circ}

135^{\circ}

The angle between northwest and north is a turn of $45^{\circ}.$

The angle between north and east is equal to $90^{\circ},$ so $45+90=135^{\circ}.$

187^{\circ}

173^{\circ}

90^{\circ}

40^{\circ}

First, add the angles given.

90+97=187

Then subtract the sum from $360.$

360-187=173^{\circ}

x=25^{\circ}; \, 75^{\circ}, \, 100^{\circ}, \, 107^{\circ}, \, 172^{\circ}, \, 181^{\circ}

x=12.56^{\circ}; \, 37.68^{\circ}, \, 50.24^{\circ}, \, 44.8^{\circ}, \, 97.36,93.92^{\circ}

x=14^{\circ}; \, 42^{\circ}, \, 56^{\circ}, \, 52^{\circ}, \, 106^{\circ}, \, 104^{\circ}

x=10.3^{\circ}; \, 30.9^{\circ}, \, 41.2^{\circ}, \, 33.5^{\circ}, \, 83.8^{\circ}, \, 78.1^{\circ}

First, solve for $x.$

Angles point 18 US

After you have found the measure of $x,$ solve for each angle.

\begin{aligned}& <A=7 x+6 \\\\ & <A=7(14)+6 \\\\ & <A=104^{\circ} \end{aligned}

\begin{aligned}& <B=3 x \\\\ & <B=3(14) \\\\ & <B=42^{\circ} \end{aligned}

\begin{aligned}& <C=4 x \\\\ & <C=4(14) \\\\ & <C=56^{\circ} \end{aligned}

\begin{aligned}& <D=5 x-18 \\\\ & <D=5(14)-18 \\\\ &<D=70-18 \\\\ & <D=52^{\circ} \end{aligned}

\begin{aligned}& <E=6 x+22 \\\\ & <E=6(14)+22 \\\\ & <E=84+22 \\\\ & <E=106^{\circ} \end{aligned}

x=345^{\circ}

x=165^{\circ}

x=15^{\circ}

x=195^{\circ}

Angles point 20 US

Angles on a straight line: $180-95=85^{\circ}$

Corresponding Angles: $85^{\circ}$

Angles on a straight line: $180-100=80^{\circ}$

Angles in a triangle: $180-(85+80)=15^{\circ}$

Angles around a point: $360-15=345^{\circ}$

Angles point FAQs

What is an angle?

An angle is a measure of the space between two intersecting lines or rays that meet at a common endpoint, called the vertex. In simpler terms, it shows how much one line “turns” relative to another.

The next lessons are

Angles in parallel lines
2D shapes
Triangles
Trigonometry

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