Sum and Difference of Angles

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Sum and Difference of Angles

We can add angles and get the result of their sum, and we can also subtract them to find the difference between them.
Even if the angles don't have any numbers, we'll learn how to represent their sum or difference and arrive at the correct result.

Angle Sum

To find the sum of angles, they must have a common vertex.

Difference Between Angles

Just as we have added angles, we can also subtract one from another.

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Test yourself on sum and difference of angles!

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Given the equilateral triangle, find X

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Even if angles don't have any numbers, we'll learn how to represent their sum or difference and arrive at the correct result: the correct naming of the angle we get as a result.
Don't worry, the sum and difference of angles is not a difficult topic and mainly relies on the representation of the angles.
Don't know how to correctly mark angles? Go practice representing angles and come back with 90% success!

Let's look at the following example

We can say that:

2 angles equal to 1

BAE+EAC=BAC∡BAE+∡EAC=∡BAC

It is known that the whole is composed of the sum of its parts, and the same is true with angles.
The large angle A) is made up of the two angles it contains.
If we add the 2 angles that make up angle A), we will obtain this angle.

If we know the size of the angles, we can, with a simple mathematical operation, discover the real value of angle A).

For example, having the following:
BAC=30°∡BAC=30°

EAC=35°∡EAC=35°

and we were asked to calculate: BAC∡BAC
which is actually the large angle A) that contains the two given angles inside,
all we have to do is add the values of the given angles and find the one we were asked to discover.

We can say that:
BAC=30°+35°=65°∡BAC=30°+35°=65°


Difference Between Angles

Just as we have added angles, we can also subtract one angle from another.
Let's look at the following example:
If we know that:

BAC=65°∡BAC=65°
BAE=30°∡BAE=30°

we know that BAC=65° BAE=30°

What will be the value of EAC∡EAC?

Since angle BAC∡BAC contains the angles BAE∡BAE and EAC∡EAC and is composed only of these two,
we can subtract the given angle BAE∡BAE from the larger angle BAC∡BAC to find the angle EAC∡EAC.
That is:

EAC=6530=35∡EAC=65-30=35
EAC=35°∡EAC=35°

Remember: The whole is composed of the sum of its parts!
We can add and subtract angles that are on the same vertex without any problem.
Just pay attention to do it the right way and know how to read the names of the angles.


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Angle Sum and Difference Exercises

Exercise 1

Assignment

Calculate the value of X X

3.c - Exercises on addition and subtraction of angles Calculate the value of x

Solution

We calculate ACB \sphericalangle ACB

ACB=180111=69 \sphericalangle ACB=180-111=69

Now we calculate ABC \sphericalangle ABC

Remember that the sum of all angles in a triangle equals 180o 180^o

ABC=1806960=51 \sphericalangle ABC=180-69-60=51

Answer

51 51


Exercise 2

Assignment

Given the angles between parallel lines in the graph, what is the value of: x x ?

c.4 - Given the angles between parallel lines in the graph

Solution

X=? X=?

180o105o=75o 180^o-105^o=75^o

75o+X=110o 75^o+X=110^o /75o /-75^o

X=110o75o X=110^o -75^o

35o 35^o

Answer

35o 35^o


Do you know what the answer is?

Exercise 3

Prompt

Given the parallel lines a,b a,b

Find the angle α \alpha

3.c -Given the parallel lines a,b

Solution

We extend the vertical line to the end and label the adjacent angles β \beta and: γ \gamma with β \beta on the left and: γ \gamma on the right

Now we notice that the angle β \beta is a corresponding angle to: 90o 90^o and since adjacent angles sum up to: 180o 180^o , then the angle γ \gamma is also equal to: 90o 90^o

The remaining angle in the small triangle we created, which is also adjacent to: 120o 120^o is called δ \delta

As it is adjacent to: 120o 120^o it will be equal to: 60o 60^o since it is complementary to: 180o 180^o

Now we calculate the sum of angles in the small triangle:

180=α+γ+δ 180=\alpha+\gamma+\delta

We replace with the data we know

180=α+90+60 180=\alpha+90+60

180=α+150 180=\alpha+150

We move the terms

α=180150 \alpha=180-150

α=30 \alpha=30

Answer

30 30


Exercise 4

Assignment

ABC \triangle ABC is a triangle

Based on the information, what is the size of the angle BAD \sphericalangle BAD

of value X X ?

3.c - ABC is a triangle

Solution

First, we calculate the angle B \sphericalangle B

B=1802886=66 \sphericalangle B=180-28-86=66

Now let's find the angle ADB \sphericalangle ADB

ADB=180122=58 \sphericalangle ADB=180-122=58

Now we refer to the triangle ABD \triangle ABD

BAD=1806658=56 \sphericalangle BAD=180-66-58=56

Answer

56 56


Check your understanding

Exercise 5

Assignment

Calculate the values of Y Y and X X

5.c - triangle 47,43,X, ABC

Solution

We refer to triangle ABC \triangle ABC

Let's find the angle Y Y

Y=1804790=43 \angle Y=180-47-90=43

Now we refer to triangle ACD \triangle ACD

Let's find the angle X X

X=1809043=47 \angle X=180-90-43=47

Answer

Y=43,X=47 Y=43, X=47


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