Collateral angles

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:

In the illustration you can see the two parallel lines $A$ and $B$ together with the transversal $C$ 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 $180º$.

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:

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.

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

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".

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".

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".

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

• Angle notation
• Sides, vertices, and angles
• Bisector
• Right angle
• Acute angle
• Obtuse angle
• Plane angle
• Adjacent angles
• Vertically opposite angles
• Alternate angles
• Corresponding angles
• Sum of the angles of a triangle
• Sum of the angles of a polygon
• Addition and difference of angles

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

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

Please explain your answer.

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.

Given the parallelogram $KLMN$ as shown in this illustration.

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

Calculate the other angles of the parallelogram $KLMN$.

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 $N$.

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

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

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

Then:

The angle $L$ measures $50º$.

The angle $K$ measures $130º$.

The angle $M$ measures $130º$.

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

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

What is angle $D$?

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,$AB$ and $DC$.

Having two parallel bases implies that the angles $A$ and $D$ are, in fact, collateral angles, i.e., we have a pair of angles located on the same side of the transversal (the side $AD$) that intersects two parallel lines ($AB$ and $DC$). 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 $180º$. Therefore, we have that the angle $D$ measures $180º-105º= 75º$.

Then:

The angle $D$ measures $75º$.

How many parallel lines are there in the figure?

Solution:

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

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

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

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

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

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

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

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

Then:

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

Given the parallel lines $a$, $b$.

Calculate the marked angles.

Solution:

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

$β = 18º$

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

$α + 18º = 180º$

$α = 162º$

Then:

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

Given the trapezoid, find the value of $X$.

Solution:

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

$\sphericalangle5+\sphericalangle3=180°$

$\sphericalangle5+\sphericalangle125=180°$

$\sphericalangle5=55°$

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

$\sphericalangle6=\sphericalangle2=102°$

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

$X=\sphericalangle7$

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

$\sphericalangle1+\sphericalangle5+\sphericalangle6+\sphericalangle7=360°$

Therefore:

$120° + 55° + 102° + X = 360°$

$X = 83°$

Then:

$X=83º$

Given the polygon in the figure:

Which of the straight lines are parallel to each other?

Solution:

• Between the sides "$b$" and "$g$" 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 "$b$" and "$d$", we have that the opposite angle to the angle of $70^o$ is internal alternate to the angle of $110^o$ That means they should be equal if "$b$" and "$d$" were parallel, which we can see does not occur.
• There is no data on "$b$" and "$b$" because there is no transversal line that intersects them, the same goes for the other pairs of sides.

Exercise 8:

$CE$ is parallel to $AD$

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

Solution:

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

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

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

We have the following equation:

$2X+\sphericalangle CAB+(X-10)=180º$

$\sphericalangle CAB=180º-2X-(X-10)$

$\sphericalangle CAB=190º-3X$

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

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

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

$(190º - 3X) + (190º - 3X) + (3X - 30) = 180º$

$350º - 3X = 180º$

$X=\frac{170º}{3}$

$X = 56.67º$

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 $180º$.