Adjacent angles

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.

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:

What can we learn from this illustration? The straight lines $A$ and $B$ 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 $C$.

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.

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 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:

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:

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:

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:

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.

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 $180º$.
  
• 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

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

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

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

Solution:

Looking at the illustration we can notice that the angle opposite to angle $B$, measures $130º$. 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 $180º$.

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

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

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 $180º$ obtaining $180º - 55º - 50º = 75º$ degrees.

Answer:

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

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

Solution:

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

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

Answer:

The angle $X$ equals $55º$.

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

And the angles marked with the letter $Y$?

Answer the question given that $ABCD$ is a rectangle.

Solution:

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

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

Answer:

Corresponding / Adjacent.

The straight lines $a$, $b$ are parallel.

Find the value of $X$.

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:

$34° + 5X + 18° = 180°$

$5X = 128°$

$X = 25.6°$

Answer:

$X=25.6°$

How many parallel lines are there in the figure?

Solution:

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

Assuming that both straight 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 adjacent angle of the angle of "$58º$" is the angle that supplements this angle. $123º$ is the angle that supplements the latter, i.e., an angle measuring $57º$ and being corresponding to the angle of $58º$ should be equal, which is clearly not true. 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. Comparing the angle adjacent to $57º$ formed by the line "$c$", which should measure $123º$ and the angle of $123º$ formed by the line "$b$", we see that they are equal as expected because they are corresponding angles. Therefore the lines "$b$" and "$c$" are parallel.

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

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

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

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

Answer:

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

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 $180º$.

What is the main characteristic of adjacent angles?

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

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 $180º$ and the sum of two obtuse angles (angles greater than $90º$) exceeds $180º$.

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

• Angle notation
• Sides, vertices, and angles
• Bisector
• Right angles
• Angles opposite at the vertex
• Alternate angles
• Corresponding angles
• Sum of the angles of a triangle
• Sum of the angles of a polygon

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