Collateral angles

🏆Practice angles in parallel lines

What are collateral angles?

The collateral angles are a pair of angles that we can find on the same side of a transversal or secant line that intersects two parallel lines, and that are also internal or external with respect to the parallel lines. The sum of the collateral angles equals180º 180º .

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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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Before we touch on the subject of collateral angles, let's see the circumstances that could lead us to use them. Let's imagine two parallel lines and a transversal line (for a more detailed explanation you should take a look at the article "Parallel lines" where this subject is discussed in depth), as we can see in the following illustration:

two parallel lines

In the illustration you can see the two parallel lines A A and B B together with the transversal C C that intersects them. In these circumstances it is very common to find collateral angles.

Now we will define the collateral angles in a more precise way that will help us to recognize them with greater certainty:

The collateral angles are the pair of angles that we can find on the same side of a transversal line that intersects two parallel lines and that are either internal or external with respect to the parallel lines. The sum of the collateral angles equals 180º 180º .

In the following illustration we can see two pairs of collateral angles. We have marked the first pair in blue and the second pair in red:

External collateral angles Internal collateral angles

A10  - External and internal collateral angles

Note that if the pair of collateral angles are outside the parallel lines then they are called external collateral angles, and if they are inside the parallel lines they are called internal collateral angles.


We still have more types of angles in the pipeline

The illustration above leads us to other types of angles. In the following, we will briefly describe a few:

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Alternate angles

Alternate angles are a pair of angles that can be found on opposite sides of a transversal that intersects two parallel lines. These angles are also on opposite sides of the parallel line to which they belong, and they are equal. If you need more detailed information, you can have a look at the article specially dedicated to this subject "Alternate angles".

Alternate angles


Corresponding angles

The corresponding angles are a pair of angles that we can find on the same side of a transversal that intersects two parallel lines. These angles are on the same side of the parallel line to which they belong and are equal.

If you need more detailed information you can have a look at the article specially dedicated to this subject "Corresponding angles".

Corresponding angles


Do you know what the answer is?

Opposite angles

Opposite angles arise at the point of intersection of two intersecting straight lines, facing each other, sharing the same vertex. Opposite angles are equal. If you need more detailed information you can have a look at the article specially dedicated to this subject " Vertically opposite angles".

C - Opposite angles


Examples with collateral angles

Exercise 1

You have to determine, in each of the illustrations below, if they are collateral angles or not.

Please explain your answer.

Illustration 2

Illustration 3

Solution:

Illustration No 1

In this illustration we indeed have a pair of collateral angles, since the two criteria that describe them are met. First, there is a pair of angles located on the same side of the transversal that intersects two parallel lines. Secondly, these angles are at opposite sides of the parallel line to which they belong.

Illustration No 2

In this illustration we are not dealing with collateral angles since one of the two criteria that describes them is not met, that is to say, we do have a pair of angles located on the same side of the transversal that intersects the two parallel lines. But, these angles are on the same side of the parallel line to which they belong.

Illustration No 3

In this illustration we are not dealing with collateral angles since one of the two criteria that describe them is not met, that is to say, there is a pair of angles that are at opposite sides of the parallel line to which they belong. However, these angles are located on different sides of the transversal that intersects the two parallel lines.

So:

Illustration No 1: collateral angles.

Illustration No 2: they are not collateral angles, however, they are corresponding angles.

Illustration No 3: they are not collateral angles, however, they are alternate angles.


Check your understanding

Exercise 2

Exercise 2 parallelogram KLMN as shown in this illustration

Given the parallelogram KLMN KLMN as shown in this illustration.

The angle N N of the parallelogram measures 50º 50º .

Calculate the other angles of the parallelogram KLMN KLMN .

Solution:

Since we are shown a parallelogram, we can take advantage of its properties to calculate the rest of the angles based on the angle N N .

In a parallelogram the opposite angles are equal, therefore, from this we can deduce that also the angle L L measures 50º 50º .

Now let's move on to the other two angles. The opposite sides of a parallelogram are parallel, i.e., side KL KL is parallel to the side NM NM . It follows that the angles K K and N N are, in fact, a pair of collateral angles, i.e., they are a pair of angles located on the same side of the transversal (KN KN ) that intersects two parallel lines (KL KL and NM NM ). Secondly, these angles are at opposite sides of the parallel line to which they belong. The collateral angles are supplementary, i.e., together they equal 180º 180º . Therefore we have that the angle K K measures 180º50º=130º 180º-50º=130º .

The angles K K and M M are also opposites within a parallelogram, so they are equal, i.e. the angle M M also measures 130º 130º .

Then:

The angle L L measures 50º 50º .

The angle K K measures 130º 130º .

The angle M M measures 130º 130º .


Exercise 3

Exercise 3 Given the rectangular trapezoid ABCD

Given the right trapezoid ABCD ABCD as described in the illustration.

The angle A A of the trapezoid measures 105º 105º .

What is angle D D ?

Solution:

We have a right trapezoid. This data is very important to solve our exercise.

Why is it important? Since it is a trapezoid, we have two parallel bases, that is,AB AB and DC DC .

Having two parallel bases implies that the angles A A and D D are, in fact, collateral angles, i.e., we have a pair of angles located on the same side of the transversal (the side AD AD ) that intersects two parallel lines (AB AB and DC DC ). Moreover, these angles are at the opposite level with respect to the parallel line to which they belong. As we have learned, the collateral angles are supplementary, that is, together they equal 180º 180º . Therefore, we have that the angle D D measures 180º105º=75º 180º-105º= 75º .

Then:

The angle D D measures 75º 75º .


Do you think you will be able to solve it?

Exercise 4

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 straight lines "aa" and "bb" are parallel:

Assuming that both lines are parallel, we have a transversal that intersects them, forming an angle of 58º 58º at the intersection with the line "aa" and an angle of 123º 123º at the intersection with the line "bb". Then, the opposite angle to the angle of 123º 123º together with the angle of 58º 58º should form a pair of collateral angles, i.e., together they should add up to 180º 180º . Remembering that opposite angles are equal, we see that the sum does not add up to 180º 180º because 57º+123º=180º 57º + 123º = 180º . Therefore, the lines "aa" and "bb" are not parallel.

Now let's see if the lines "bb" and "cc" are parallel:

Using similar reasoning as above, we can assume that these lines are parallel. Then using the opposite angle to the angle of 57º 57º formed by the straight line "cc", and the angle of 123º 123º formed by the straight line "bb", we see that they are internal collaterals, so their sum should be 180º 180º . In this case this is true since 57º+123º=180º 57º + 123º = 180º therefore the lines "bb" and "cc" are parallel.

Now let's see if the lines "cc" and "dd" are parallel:

The angle of 57º 57º formed by the line "cc" and the angle of 123º 123º formed by the line "dd" are internal collateral angles, so their sum should be 180º 180º . In this case this is true since 57º+123º=180º 57º + 123º = 180º therefore the lines "cc" and "dd" are parallel.

Now let's see if the lines "dd" and "ee" are parallel:

Again, let's assume that they are. The angle of 30º 30º formed by the line "dd" and the angle of 150º 150º formed by the line "ee" form external collateral angles, therefore, toether they should equal 180º 180º . This is true because 30º+150º=180º 30º + 150º = 180º Therefore, the lines "dd" and "ee" are parallel.

Then:

The straight lines " b,c,d,e“b”, “c”, “d”, “e” " are parallel.


Exercise 5

Given the parallel lines aa, bb.

Calculate the marked angles.

Given the parallel lines a, b Calculate the marked angles

Solution:

Angle β β is opposite to the angle that equals 18º 18º .

β=18º β = 18º

The angle of 18º 18º happens to be internal collateral with angle α, so their sum must equal 180º 180º :

α+18º=180º α + 18º = 180º

α=162º α = 162º

Then:

The angles are β=18º β = 18º y α=162º α = 162º .


Test your knowledge

Exercise 6

Given the trapezoid, find the value of X X .

Given the trapezoid, find the value of new X

Solution:

Since the sides DC DC y AB AB are parallel, the angles formed by the transversal lines will be:

5+3=180° \sphericalangle5+\sphericalangle3=180°

5+125=180° \sphericalangle5+\sphericalangle125=180°

5=55° \sphericalangle5=55°

The angles \( \sphericalangle2 \) and \( \sphericalangle6 \) are opposite angles, so they are equal:

6=2=102° \sphericalangle6=\sphericalangle2=102°

In the same way, the angles X X and 7 \sphericalangle7 are opposite angles, so they are equal.

X=7 X=\sphericalangle7

Then, remembering that the sum of the angles of a quadrilateral is equal to 360º 360º we have that:

1+5+6+7=360° \sphericalangle1+\sphericalangle5+\sphericalangle6+\sphericalangle7=360°

Therefore:

120°+55°+102°+X=360° 120° + 55° + 102° + X = 360°

X=83° X = 83°

Solution to the previous exercise

Then:

X=83º X=83º


Exercise 7

Given the polygon in the figure:

Exercise 7 Given the polygon of the figure

Which of the straight lines are parallel to each other?

Solution:

  • Between the sides "bb" and "gg" we can identify a pair of internal alternate angles that are not equal, therefore those sides are not parallel.
  • Considering the angles formed by the transversal that intersects the sides "bb" and "dd", we have that the opposite angle to the angle of 70o 70^o is internal alternate to the angle of 110o 110^o That means they should be equal if "bb" and "dd" were parallel, which we can see does not occur.
  • There is no data on "bb" and "bb" because there is no transversal line that intersects them, the same goes for the other pairs of sides.

Exercise 8:

CE CE is parallel to AD AD

Exercise 8 CE is parallel to AD

What is the value of X X if it is given that ABC ABC is an isosceles triangle, with sides AB=BC AB=BC

Solution:

The angle ACE \sphericalangle ACE is opposite to the angle2X 2X therefore they are equal.

By the same argument the angle DAB \sphericalangle DAB is equal to X10 X - 10 .

Then, the sum of the internal alternate angle to the angle ACE \sphericalangle ACE together with the angle CAB \sphericalangle CAB and the angle BAD \sphericalangle BAD add up to 180º 180º

We have the following equation:

2X+CAB+(X10)=180º 2X+\sphericalangle CAB+(X-10)=180º

CAB=180º2X(X10) \sphericalangle CAB=180º-2X-(X-10)

CAB=190º3X \sphericalangle CAB=190º-3X

Now, since the sides AB=BC AB = BC then the angles ACB=CAB=190º3X \sphericalangle ACB=\sphericalangle CAB=190º-3X .

Next, recall that the sum of the interior angles of a triangle is 180º 180º therefore:

ACB+CAB+ABC=180º \sphericalangle ACB+\sphericalangle CAB+\sphericalangle ABC=180º

(190º3X)+(190º3X)+(3X30)=180º (190º - 3X) + (190º - 3X) + (3X - 30) = 180º

350º3X=180º 350º - 3X = 180º

X=170º3 X=\frac{170º}{3}

X=56.67º X = 56.67º


Do you know what the answer is?

Questions on the subject

What are collateral angles?

They are a pair of angles that we can find on the same side of a transversal or secant line that intersects two parallel lines and that are either internal or external with respect to the parallel lines.


Where do the collateral angles appear?

When two parallel lines are intersected by a transversal or secant line.


What is the sum of a pair of collateral angles?

Being supplementary angles, their sum is 180º180º.


Check your understanding

examples with solutions for collateral 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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