Adjacent angles

🏆Practice angles in parallel lines

What does adjacent angle mean?

Adjacent angles are the pair of angles formed when two lines intersect each other. These angles are formed at the point where the intersection occurs, and are adjacent to eachother - hence its name. Another pair of angles that are formed at the intersection of two straight lines are the opposite angles, but this pair of angles are opposite at the vertex and not adjacent, so we should not confuse them with adjacent angles. Adjacent angles are always supplementary, that is, together they equal 180° 180° .

The following illustration shows two examples of what adjacent angles look like. One example is red and the other blue.

Adjacent angles new

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Test yourself on angles in parallel lines!

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If one of two corresponding angles is a right angle, then the other angle will also be a right angle.

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When we are faced with verbal mathematical problems in geometry we must use the full arsenal we have, that is, analyze the different types of angles that arise when straight lines are parallel or intersect. Knowing and understanding the different types of angles, along with our ability to use the various properties of each, can greatly facilitate the resolution of many geometry problems.

In this section we will focus on adjacent angles, but we will briefly review their "siblings," corresponding angles, alternate angles, angles opposite at the vertex and collateral angles.

What are adjacent angles?

Before elaborating on the subject of adjacent angles, we will begin by explaining the situation that allows for the formation of these angles. To simplify, we we will be guided by the illustration of two parallel lines and a transversal, as shown in the following:

A2 - Parallel lines

What can we learn from this illustration? The straight lines A A and B B are parallel (although in our case we don't need parallel lines to find adjacent angles), and these are intersected by a transversal or secant line C C .

Now, after having reviewed some basic theory with the help of a graphic example, we will move on to the subject that interests us, that is to say, we will learn the definition of adjacent angles, which in turn will help us to identify them more easily.


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Different types of angles.

As we mentioned at the beginning of this section, adjacent angles are not the only angles that we can have in parallel lines. Next, we will briefly review other types of angles that can help us to solve geometric exercises:

Corresponding angles

Corresponding angles arise when a transversal crosses two or more parallel straight lines. The corresponding angles are on the same side of the transversal and at the same side of their parallel lines. Corresponding angles are equal.

If you need further explanation, you can refer to the article "Corresponding angles".

The following illustration represents a pair of corresponding angles in black:

Corresponding angles


Do you know what the answer is?

Alternate angles

Alternate angles arise when a transversal crosses two or more parallel lines. The alternate angles are on the opposite side of the transversal and on different sides of the parallel lines. The alternate angles are equal.

If you need further explanation, you can refer to the article "Alternate angles".

The following diagram shows two pairs of alternate angles, the first pair in red and the second pair in blue:

3.c - Alternate angles new


Collateral angles

Collateral angles arise when a transversal crosses two or more parallel lines. The collateral angles are on the same side of the transversal and at on different sides of the adjacent parallel lines. The collateral angles supplement eachother, that is, together they equal 180º.

If you need a deeper explanation, you can refer to the article "Collateral angles".

The following illustration represents a pair of these angles in red:

A10  - External and internal collateral angles


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Opposite Angles

Opposite angles arise at the point of intersection of two straight lines. These angles are opposite at the vertex and are equal.

If you need a deeper explanation, you can visit the article "Angles opposite at the vertex ".

The following diagram represents two pairs of these angles, both in red:

O - Angles opposite


Practice exercises

Exercise 1

In each of the following illustrations, determine if the angles are adjacent. If yes, explain why. If the answer is no, explain why and specify what type of angle is seen in each illustration.

1 -  Illustration No 1

2 Illustration No 2

3 Illustration No 3

Solution:

  • Illustration No 1
    In this diagram there are no adjacent angles, rather they are opposite angles. According to its definition, opposite angles are two angles that arise at the point of intersection of two straight lines, are equal and are opposite each other.
  • Illustration No 2
    In this picture we are effectively dealing with adjacent angles. The reason is that, according to its definition, adjacent angles are formed when two straight lines intersect each other, and are formed at the point where the intersection occurs, one adjacent to the other, a characteristic from which derives its name. Adjacent angles are supplementary, that is, together they equal 180º 180º .
  • Illustration No 3
    In this scheme there are no adjacent angles, rather they are corresponding angles. According to its definition, corresponding angles are two angles that arise when a transversal crosses two or more parallel straight lines, are on the same side of the transversal, and on the same side of the adjacent parallel lines. The corresponding angles are equal.

Answer:

Illustration No 1: Opposite angles.

Illustration No 2: Adjacent angles

Illustration No 3: Corresponding angles


Do you think you will be able to solve it?

Exercise 2

In the triangle ABC ABC , the sides are extended to show some of its exterior angles.

The angle at the vertex A A of the triangle measures 55º 55º .

Calculate the other two angles based on this data and those shown in the illustration.

Adjacent_angles_-_Exercise_02.original

Solution:

Looking at the illustration we can notice that the angle opposite to angle B B , measures 130º 130º . These angles are opposite angles and are equal. According to its definition, adjacent angles are those formed when two straight lines intersect each other, are formed at the point where the intersection occurs, one adjacent to the other, a characteristic from which it gets its name. Adjacent angles always supplement each other, that is, together they equal 180º 180º .

It follows that the angle of the vertex B B of the triangle ABC ABC equals 180º130º=50º 180º - 130º = 50º .

Then, as is well known, the sum of the interior angles of a triangle is equal to 180º 180º .

Since we know the measure of two of the three angles of the triangle, the remaining angle corresponding to vertex C is calculated by subtracting the two angles from 180º 180º obtaining 180º55º50º=75º 180º - 55º - 50º = 75º degrees.

Answer:

The angle B B and C C of the triangle ABC ABC equal 50º 50º and 75º 75º respectively.


Exercise 3

Calculate the angle X X based on the data given in the illustration and the material learned.

Adjacent_angles_-_Exercise_03.original

Solution:

We notice from the illustration that we are dealing with three angles that together create a straight angle. A straight angle is 180º 180º . The angle X X is adjacent to the right angle (90º 90º ) and another one that equals 35º 35º , that is to say the two together measure 125º 125º . The angle X X and the angle measuring 125º 125º are adjacent angles and supplement each other, that is, together they equal 180º 180º .

From the above we obtain that the angle X X equals 180º125º=55º 180º - 125º = 55º degrees.

Answer:

The angle X X equals 55º 55º .


Test your knowledge

Exercise 4

What are the angles marked with the letter X in the figure?

Find the size of the angles marked with the letter X X ?

And the angles marked with the letter Y Y ?

Answer the question given that ABCD ABCD is a rectangle.

Solution:

Since ABCD ABCD is a rectangle, the side AB AB and DC DC are parallel.

Then, the angles marked with the letter X X are corresponding because they are on the same side of the transversal that forms them and on the same side of their adjacent parallel lines.

Finally, the angles marked with the letter X X are adjacent since they are formed at the intersection of the extension of the sides AD AD and DC DC of the rectangle.

Answer:

Corresponding / Adjacent.


Exercise 5

The straight lines a a , b b are parallel.

Find the value of X X .

Lines a,b are parallel

Solution:

Taking into account the following illustration, we have that angle 1 is corresponding to angle 3, therefore they are equal.

Then, angle 1 and angle 2 are adjacent angles, therefore they are supplementary and that gives us the following equation:

34°+5X+18°=180° 34° + 5X + 18° = 180°

5X=128° 5X = 128°

X=25.6° X = 25.6°

Solution to Exercise 5

Answer:

X=25.6° X=25.6°


Do you know what the answer is?

Exercise 6

Exercise 4 How many parallel lines are there in the figure?

How many parallel lines are there in the figure?

Solution:

How many parallel lines are there in the figure in front of you?

Let's start by seeing if the lines "a a " and "b b "are parallel:

Assuming that both straight lines are parallel, we have a transversal that intersects them, forming an angle of 58º 58º at the intersection with the line "a a " and an angle of 123º 123º at the intersection with the line "b b ". Then, the adjacent angle of the angle of "58º 58º " is the angle that supplements this angle. 123º 123º is the angle that supplements the latter, i.e., an angle measuring 57º 57º and being corresponding to the angle of 58º 58º should be equal, which is clearly not true. Therefore, the lines "a a " and "b b " are not parallel.

Now let's see if the lines "b b " and "c c " are parallel:

Using similar reasoning as above, we can assume that these lines are parallel. Comparing the angle adjacent to 57º 57º formed by the line "c c ", which should measure 123º 123º and the angle of 123º 123º formed by the line "b b ", we see that they are equal as expected because they are corresponding angles. Therefore the lines "b b " and "c c " are parallel.

Now let's see if the lines "c c " and "d d " are parallel:

The angle adjacent to the angle of 57º 57º contained in the line "c c " which is an angle of 123º 123º and the angle of 123º 123º contained in the line "d d ", are corresponding angles. In this case they are indeed equal, therefore the straight lines "c c " and "d d " are parallel.

Now let's see if the lines "d d " and "e e " are parallel:

Again let us suppose that they are. Then, the angle adjacent to 30º 30º formed by the line "d d " measures 150º 150º and together with the angle of 150º 150º formed by the line "e e " form corresponding angles, therefore, they must be equal. This is true, therefore the lines "d d " and "e e " are parallel.

Answer:

The lines "b b ", "c c ", "d d ", "e e " are parallel.


Review questions

What are adjacent angles?

They are a pair of angles of angles formed by the intersection of two straight lines in such a way that they supplement each other, that is to say, that together they form 180º 180º .


What is the main characteristic of adjacent angles?

That they are supplementary, that is, when added together they form a straight angle (180º 180º ).


In a diagram of parallel lines intersected by a transversal, which pair of angles have the same property of adjacent angles, that is, that they are supplementary?

The collateral angles.


In a pair of adjacent angles, can either one separately be an obtuse angle?

No, since being adjacent angles they are supplementary, i.e., together they add up to 180º 180º and the sum of two obtuse angles (angles greater than 90º 90º ) exceeds 180º 180º .


If you are interested in learning more about angles, you can access one of the following articles:

On the Tutorela you will find a variety of articles about mathematics.


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examples with solutions for adjacent angles

Exercise #1

Which type of angles are shown in the figure below?

Step-by-Step Solution

Alternate angles are a pair of angles that can be found on the opposite side of a line that cuts two parallel lines.

Furthermore, these angles are located on the opposite level of the corresponding line that they belong to.

Answer

Alternate

Exercise #2

a a is parallel to

b b

Determine which of the statements is correct.

αααβββγγγδδδaaabbb

Video Solution

Step-by-Step Solution

Let's review the definition of adjacent angles:

Adjacent angles are angles formed where there are two straight lines that intersect. These angles are formed at the point where the intersection occurs, one next to the other, and hence their name.

Now let's review the definition of collateral angles:

Two angles formed when two or more parallel lines are intersected by a third line. The collateral angles are on the same side of the intersecting line and even are at different heights in relation to the parallel line to which they are adjacent.

Therefore, answer C is correct for this definition.

Answer

β,γ \beta,\gamma Colateralesγ,δ \gamma,\delta Adjacent

Exercise #3

Is it possible to have two adjacent angles, one of which is obtuse and the other right?

Video Solution

Step-by-Step Solution

Remember the definition of adjacent angles:

Adjacent angles always complement each other up to one hundred eighty degrees, that is, their sum is 180 degrees.

This situation is impossible since a right angle equals 90 degrees, an obtuse angle is greater than 90 degrees.

Therefore, together their sum will be greater than 180 degrees.

Answer

No

Exercise #4

In which of the diagrams are the angles α,β  \alpha,\beta\text{ } vertically opposite?

Step-by-Step Solution

Remember the definition of angles opposite by the vertex:

Angles opposite by the vertex are angles whose formation is possible when two lines cross, and they are formed at the point of intersection, one facing the other. The acute angles are equal in size.

The drawing in answer A corresponds to this definition.

Answer

αααβββ

Exercise #5

The lines a and b are parallel.

What are the corresponding angles?

αααβββγγγδδδaaabbb

Video Solution

Step-by-Step Solution

Given that line a is parallel to line b, let's remember the definition of corresponding angles between parallel lines:

Corresponding angles are angles located on the same side of the line that intersects the two parallels and are also situated at the same level with respect to the parallel line to which they are adjacent.

Corresponding angles are equal in size.

According to this definition α=β \alpha=\beta and therefore the corresponding angles

Answer

α,β \alpha,\beta

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