Angle Addition Postulate: Explained with Examples · Matter of Math (2024)

This lesson will give you the definition of the angle addition postulate, visual examples, and explanations and how it is used.

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

The Angle Addition Postulate: A Definition
Actual Meaning: The Main Idea
Real-Life Application
Another Postulate: The Segment Addition Postulate
Geometry Practice Questions
Refresher: Parts of the Angle
To Sum Up (Pun Intended!)

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

Actual Meaning: The Main Idea

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.

Angle Addition Postulate: Explained with Examples · Matter of Math (1)

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°.

Angle Addition Postulate: Explained with Examples · Matter of Math (2)

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.

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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 measure of the 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.

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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.

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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:

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

AC = AB + AC
28cm = 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 · 5cm = 10cm

Geometry 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!

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Using the postulate, form the equation

∠AOC = ∠AOB + ∠BOC
= 32° + 42°
= 74°

Angle ∠AOC is 74°.

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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° + 62°.
∠DEH = 173°

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

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

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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°

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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

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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

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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.

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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 + 25
180 = 3x + 30
150 = 3x
x = 50

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

∠CAD = 50+25
∠CAD = 75°

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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°

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∠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

Parts of The Angle: A Brief Refresher

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.

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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:

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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.

To Sum Up (Pun Intended!)

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

This can be applied similarly to finding the sums of line segment lenghts.

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

For more help and lessons, head to the homepage.

For now, hopefully, you 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!

Angle Addition Postulate: Explained with Examples · Matter of Math (2024)

FAQs

Angle Addition Postulate: Explained with Examples · Matter of Math? ›

The Angle Addition Postulate states that the sum of two adjacent angle measures will equal the angle measure of the larger angle that they form together. The formula for the postulate is that if D is in the interior of ∠ ABC then ∠ ABD + ∠ DBC = ∠ ABC. Adjacent angles are two angles that share a common ray.

How do you explain arc addition postulate? ›

The measure of an arc formed by two adjacent arcs in a circle is called an arc addition postulate. The measure of the large arc created is equal to the sum of the two adjacent arcs.

What is the addition postulate theorem? ›

Addition Postulate If equal quantities are added to equal quantities, the sums are equal. Transitive Property If a = b and b = c, then a = c.

What is the linear pair postulate and angle addition postulate? ›

Angle Addition Postulate: If P is in the interior of ∠RST , then m∠RSP + m∠PST = m∠RST . Linear Pair Postulate: If two angles form a linear pair, then they are supplementary. Right Angle Congruence Theorem: All right angles are congruent.

What is postulate 5 and examples? ›

That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

What is the rule of arcs and angles? ›

The angle of an arc is identified by its two endpoints, written as mAB. The measure of an arc angle is found by dividing the arc length by the circle's circumference, then multiplying by 360 degrees. Formulas for calculating arcs and angles vary based on where they are in reference to the circle.

What is the mathematical meaning of arc? ›

In Mathematics, an “arc” is a smooth curve joining two endpoints. In general, an arc is one of the portions of a circle. It is basically a part of the circumference of a circle. Arc is a part of a curve. An arc can be a portion of some other curved shapes like an ellipse but mostly refers to a circle.

How do you describe an inscribed angle? ›

In geometry, an inscribed angle is the angle formed in the interior of a circle when two chords intersect on the circle. It can also be defined as the angle subtended at a point on the circle by two given points on the circle.

What is an example of the addition theorem? ›

Example: The event of getting a head and the event of getting a tail when a coin is tossed are mutually exhaustive. Addition theorem on probability: If A and B are any two events then the probability of happening of at least one of the events is defined as P(AUB) = P(A) + P(B)- P(A∩B).

Does a postulate need to be proven? ›

Answer and Explanation:

No, a postulate does not require proof. A more technical definition of a postulate in math is a statement that is generally accepted as true with or without a proof indicating as such. By this definition, we see that postulates are accepted as true whether they have a proof or not.

How to explain angle addition postulate? ›

The Angle Addition Postulate states that the sum of two adjacent angle measures will be equal to the measure of the larger angle they form. The postulate can also be used to find the measure of one of the smaller angles if the larger angle and one adjacent angle measure is provided.

What is the symbol for an angle? ›

The symbol ∠ is used to denote an angle. The symbol m ∠ is sometimes used to denote the measure of an angle. An angle can be named in various ways (Figure 2). Figure 2 Different names for the same angle.

What is the angle angle postulate? ›

Angle-Side-Angle (ASA) Postulate

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

What is an example of a postulate and theorem? ›

If two planes intersect, then their intersection is a line (Postulate 6). A line contains at least two points (Postulate 1). If two lines intersect, then exactly one plane contains both lines (Theorem 3). If a point lies outside a line, then exactly one plane contains both the line and the point (Theorem 2).

What is an example of SSS congruence postulate? ›

What is an example of the SSS postulate or theorem? The SSS postulate applies to triangles that have the same measurements for corresponding sides. An example would be a triangle that has side lengths 3, 4, and 5 and a triangle that has side lengths 4, 3, and 5.

References

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