Angle Addition Postulate: Explained with Examples

This lesson will guide you through the different types of angles, the definition of the angle addition postulate, and its uses.

Angles can be found everywhere – the hands of a clock, wheels, pyramids and most importantly in design and construction of architecture, such as roads and buildings.

Once you’re confident in the basics of angles and how the postulate works, you will be able to work through the practice questions at the end of this lesson.

Contents

Parts of the Angle
Angle Types
The Angle Addition Postulate: A Definition
What Does That Actually Mean?
Real-Life Application
Another Postulate: The Segment Addition Postulate
Practice Questions
To Sum Up (Pun Intended!)

Parts of The Angle

An angle is formed when two lines or rays meet at the same endpoint.

The symbol ∠ can be used to represent angles. The angle below is written ∠ABC.

Angle ABC

Angles are usually measured in degrees, which are represented by the symbol °. We would write the name and size of the angle above like this:

∠ABC = 60°

BA and BC are the sides of the angle, also known as rays.

B is the common vertex – the point they share between the sides BA and BC.

Important: when naming an angle, the middle letter must be the common vertex.

The interior angle is the angle between the two sides, whereas the exterior angle is the angle outside of the two sides.

The last definition you need before moving on is for adjacent angles, which share a side and a vertex.

Here is an example:

Adjacent angles

See how the angles share the vertex, O, and the line in the middle, OB.

The angle x can be shown as ∠AOB.


Angle y is ∠BOC or ∠COB

Angle z is ∠AOC or ∠COA.

As you can see, it doesn’t matter which order you put the letters in, as long as the common vertex is in the middle, “O” in the case above.

Angle Types

Types of angles acute right reflex straight obtuse reflex

The type of angle depends on its size. In order of size, there are:

  1. Acute – 0° < angle < 90°
  2. Right – Exactly 90°
  3. Obtuse – 90° < angle < 180°
  4. Straight – Exactly 180°
  5. Reflex – 180° < angle < 360°

Reflex angles are the exterior angles.

They are usually given as the acute or obtuse angle – the interior of the same two rays – as these are easier to find using a standard protractor.

Acute Obtuse Interior, Reflex Interior

You can find the reflex angle by subtracting the acute angle from 360°, which is the angle of a full circle.

Reflex = 360° – Acute
OR
Reflex = 360° – Obtuse

Get really comfortable with the different types of angles – you should be able to name them all without thinking.

The Angle Addition Postulate: A Definition

The textbook definition goes a little like this:

If the point B lies in the interior of angle AOC then

∠AOB + ∠BOC = ∠AOC

What Does That Actually Mean?

So, if you place two angles side by side, they are adjacent. Then the new angle made by both together is the sum of the two original angles.

You can picture this using two arrowheads.

Green Blue Arrows

The blue arrowhead has sides BL and UL, so the vertex is L. The tip of the arrow forms the angle ∠BLU which measures 40°.

The green arrowhead has sides GR and NR, so the vertex is R. These three points create ∠GRN which measures 60°.

Green Blue Arrows Added

By placing the two arrowheads side by side so that the points L and R join, and the points U and G join, a pair of adjacent angles has been made.

Brown arrow

This has created a new angle measured from side B to N. This is angle ∠BRN.

By adding the two adjacent angles ∠GRN and ∠BLU together, you can find ∠BRN.

So in this case…

∠BRN = ∠BLU + ∠GRN 
  = 40° + 60°
= 100°      

So, there you have it! The ∠BRN is 100°.

Here’s a fun tool to play around with and explore how changing the size of two adjacent angles affects the size of their resulting angle.

You will find that changing points A, D, or C will affect the resulting angle it makes, without affecting the adjacent angle.

However— notice how the resulting angle changes? This is because it is the sum of the two adjacent angles.

Now you know how the postulate works, let’s work through an example and calculate the resulting angle.

Adding JKL and JKM

As you can see these angles share the same side KL, so they are adjacent.

The angle ∠JKL is a right angle so it is 90°, and from the diagram, you will see LKM is 30°.

You can find their resulting angle as the sum of 90° and 30° so ∠JKM is 120°.

Real-Life Application: Angle Addition Postulate

Now you know how the postulate works, you must know how it can be used in real life.

There are many applications of the postulate, especially in architecture and engineering.

Roof trusses are beams of timber organized in triangles in the roofs of buildings.

It is important the angles in each triangle are measured correctly, as roof trusses provide support for a roof.

The Howe truss is made up of two 60° triangles and the Fink truss is made with three 40° triangles.

Roof Truss

The same idea also applies to bridges. Some bridges have cables connected to bridges at angles from the bridge floor to towers.

These cables placed at specific angles support the bridge’s structure by sharing the weight of the bridge evenly across its supports.

Another Postulate: The Segment Addition

The Segment Addition Postulate is similar to the angle addition postulate, but you are working with line segments instead of adjacent angles.

If the point B is between A and C on a line segment, then:

AC = AB + BC

To keep it simple, you can add connected line segments in the same way you can add adjacent angles!

Thank you to Lamee Storage for the video.

Here’s a worked example:

Segment Addition Postulate Example

Use the postulate, substitute the values that we know, and do a little rearranging:

    AC = AB + AC
 28 = 5x + 3
28 – 3 = 5x           
    x = 25 ÷ 5

Now you have found x, substitute this into the formula for AB which is 2x.

AB = 2 · 5 = 10

Practice Questions

Please don’t try and use a protractor to find the angles. Not only will you miss out on the valuable practice, but you’ll get the answer wrong… because they’re not drawn accurately!

Practice 1


Using the postulate, form the equation

∠AOC = ∠AOB + ∠BOC
  = 32° + 44°
= 76°        

Angle ∠AOC is 76°.

Practice 2


Write out the postulate.

∠DEH = ∠DEF + ∠FEG + ∠GEH

Using the fact that ∠DEF is a right angle, calculate the sum of the 3 adjacent angles.

     ∠DEH = 90° + 21° + 64°.
∠DEH = 175°              

Practice 3


The angle ∠JKM is straight, so the two adjacent angles sum to 180°.

     ∠LKM = ∠JKM – ∠JKL.
∠LKM = 180° – 48°
∠LKM = 132°          

Practice 4


Angle ∠MOP is a right angle, so the two adjacent angles add up to 90°.

To find ∠MON subtract ∠NOP from 90°.

   ∠MON = 90° – 12°
∠MON = 78°      

Practice 5


The sum of angle ∠RQS and ∠SQT is equal to 136°.

The sum of these adjacent angles

8x – 4 + 4x + 20 = 12x + 16

Solve for x using the size of ∠RQT.

     136 = 12x + 16
120 = 12x    
x = 10  

Practice 6


Find the sum of ∠VUW and ∠WUX to find the angle ∠VUX.

∠WUX is a right angle so it is 90° and ∠VUW is 48° so their sum is 138°.

Solve this with the equation for ∠VUX.

10x + 8 = 138        
10x = 130 
 x = 13

Practice 7


Form the equation using the postulate.

∠XWZ = ∠XWY + ∠YWZ

The sum of the two adjacent angles is:

x + 42 + x + 77 = 2x + 119

From the question, you know the angle ∠XWZ is 95 so:

2x + 119 = 95            
2x = -24
   x = -12.

Practice 8


Angle ∠BAD is a straight line so it is 180°.

Using the formula ∠BAD=∠BAC+∠CAD, you can solve:

180 = ∠BAC + ∠CAD

Then find x.

          180 = 2x + 5 + x + 2
180 = 3x + 30
150 = 3x         
x = 50     

Substitute this value of x into the equation for ∠CAD.

∠CAD = 50+25
∠CAD = 75°     

Practive 9


Using the formula:

∠EFH = ∠EFG + ∠GFH

Find the sum of the two adjacent angles.

∠EFG + ∠GFH = 42 + 12x – 4       
            = 12x + 38

Using the equation given for ∠EFH:

 17x + 8 = 12x + 38
5x = 30 
x = 6

Substitute this value of x into the equation for ∠EFH.

∠EFH = 110°

Practice 10


∠DAE is a right angle, so it is 90°.

BE is a straight line, so ∠BAE is 180°. This means ∠BAC, ∠CAD, and ∠DAE sum to 180°.

180 = 13x + 30 + 2x + 15 + 90

Rearrange and solve for x.

        180 = 15x + 135
15x = 45     
x = 3   

To Sum Up (Pun Intended!)

In this lesson, you have learned how to classify angles into their different types – acute, right, obtuse, straight, and reflex.

You also saw how to define and recognize adjacent angles, which is important in applying the angle addition postulate.

By making two angles adjacent, you can find their resulting angle by finding the sum of the two original angles.

This can be applied similarly to finding the sums of lengths of line segments.

For more help and lessons, head to the homepage.

For now, hopefully, you now feel confident in finding the total of adjacent angles. Post your answers to any of the challenges or leave any questions in the comments below!

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