Corresponding Angles

We all must have very clear what an angle is, right? It is a geometrical figure formed by two rays that have equal initial point. Now, what comes to mind when you listen to the expression “corresponding angles”? Maybe is there an angle that corresponds to each person?

Well… That doesn’t sound very geometrical! But don’t worry, we are about to learn step by step what is the exact definition of this expression, and after learning it you will sure find many of these angles around you!

Intersecting, parallel and transversal lines

Before defining what corresponding angles are, we need to understand the difference between three types of lines.

Intersecting lines: we say that two lines l₁ and l₂  are intersecting if they cross. In other words, these two lines share a point. Like the two yello lines shown below that share point P. Point P is called point of intersection between  l₁ and l₂ .

Parallel lines: two lines  l₁ and l₂   are called parallel if they don’t intersect (cross). That is, the two lines don’t have any point in common. Like the two turquoise lines shown below.

Transversal line: a line 𝑙 is transversal if it intersects (crosses) two lines at two different points. In the image below, 𝑙 is transversal to l₁ and l₂  because it intersects both lines in P and Q, respectively, and these two points are different. However, 𝑙 is not transversal to ${\mathcal{<em><strong>l₃</strong></em> and\; <em><strong>l₄</strong></em> }}_{}$ because it intersects the two lines in the common point R.

It is important to notice the following two facts:

• In order to define a transversal line, two more lines need to be considered.
• The two lines that the transversal intersects can be of any kind (parallel or intersecting), what matters is that the transversal crosses them at different points.

Example: Which of the farm gate bars are parallel, intersecting or transversal?

We see that all 4 horizontal bars are parallel because they don’t cross each other. Each vertical bar is transversal because they intersect the horizontal ones exactly where each nail is. Also, each of the inclined bars is transversal to the parallel ones. Besides, four pairs of inclined bars are intersecting.

Did you know that…
Some leaves have parallel veins? This pattern is called “parallel venation” and can be found in the leaves of coconuts and bananas, for example.

What are corresponding angles?

With the previous concepts, we are ready to learn what corresponding angles mean. If 𝑙 is transversal to lines ${\mathcal{<em><strong>l₁</strong></em> and\; <em><strong>l₂</strong></em> }}_{}$, the points of intersection determine eight angles, as shown below.

We say that two angles are corresponding angles if they are on the same side of ${\mathcal{<em><strong>l₁</strong></em> and\; <em><strong>l₂</strong></em> }}_{}$, and on the same side of the transversal.

In the graphic above, angles 1 and 2 (the pink ones) are corresponding angles because they are both to the right of the transversal 𝑙, angle 1 is above l₁ , and angle 2 is also above l₂ .

There are three more pairs of corresponding angles: angles 3 and 4 (the blues ones), angles 5 and 6 (the yellow ones), and angles 7 and 8 (the purple ones).

According with their location, we can classify the four pairs as follows:

• angle 1 and angle 2 are upper right side angles.
• angle 3 and angle 4 are lower right side angles.
• angle 5 and angle 6 are upper left side angles.
• angle 7 and angle 8 are lower left side angles.

Example: Identify four pairs of corresponding angles in the image, and indicate the location of each pair.

Let’s imagine that each string of the racket
is part of a line, as shown in the image.

Notice that 𝑙 is transversal to lines l₂ and l₃ , that the yellow angles are upper left side angles, and the blue angles are lower right side angles respect to these three lines. Thus, the yellow angles are corresponding angles; and the blue angles are also corresponding angles.

Besides, line 𝑚 is transversal to l₁ and l₂ , and the green angles are upper right side angles with respect to these three lines, thus they are corresponding angles.

Finally, the pink angles are corresponding angles because line 𝑛 is transversal to l₂ and l₃ , and the angles are lower left side angles respect to these three lines.

There are more pairs of corresponding angles, but you can surely find them
yourself!

Corresponding angles formed by parallel lines

Did you notice something particular with the strings in the racket example?

Hint: parallel lines

Exactly! Lines ${\mathcal{and\; <em><strong>l₃</strong></em> }}_{}$ were parallel because those strings in rackets are parallel. Also, every two corresponding angles in that example seemed to have the same measure. Is that possible?

Well, it is! As we are about to learn from the following rule that is called the Corresponding Angles Postulate.

Corresponding Angles Postulate: If a transversal line intersects two parallel lines, the corresponding angles determined by them have the same measure.

Therefore, it may be the case that corresponding angles don’t measure the same, but they will whenever the transversal crosses two parallel lines. It was the case in the racket example, as we suspected!

Furthermore, the converse of the postulate is also true: if a pair of corresponding angles measure the same, then the lines intersected by the transversal are parallel.

Example: The furniture in the image has four parallel shelves, and two supports in each side, crossing the shelves.

By the postulate, we get that the corresponding angles are drawn in the image measure the same.

Specifically, the upper right side angles have the same measure, and the lower left side angles have equal measures.

Moreover, as appears in the image, the measure of the lower left side angles is greater than 90°, and that of the upper right side angles is less than 90°.

Can you tell which other pairs of angles have equal measure?

Example: The turquoise line drawn on the wooden sign of the image, is transversal to the yellow ones.

It is clear from the image that angle 2 is greater than angle 1. Thus, the yellow lines can’t be parallel. If they were parallel these two corresponding angles would measure the same.

In other words, the turquoise lines are intersecting, and they will cross somewhere if we prolong their drawings.

Challenging problems

Since you are almost an expert on corresponding angles, we now propose to you a couple of problems for you to solve using the notions that we have studied so far.

But don’t worry! We won’t leave you alone, we will give you some hints to solve these problems.

Look at the landscape below. We have measure angles 1 and 2, and we have found that they measure the same.

• What can be said about the railway sleepers?
• Is there another angle in the picture whose measure is the same as that of angle 3?
• Could you find in the landscape another pair of corresponding angles?

For question 1, notice that the rails are transversal to the railway sleepers. Then, since angle 1 and angle 2 have the same measure, the railway sleepers
have to be parallel. Why?

To answer question 2, notice that angle 3 and any of the upper left side angles above it are corresponding angles. Does this imply that the measure of those angles is the same?

For question 3, look around! Can you see some corresponding angles made by the branches of the trees, for example?

Let’s now consider the hashtag symbol from the image.
We can see 16 angles and several pairs of parallel lines in the image. Can you?

We invite you to solve the following problems:

• Find a group of four angles with the same measure in the hashtag sign.
• Find a different group of three angles with the same measure in the hashtag sign.

A hint to solve problem 1 is to first draw some parallel lines. For example, two
horizontal parallel red lines, and two vertical parallel white lines, like in the image below.

Notice that the two drawn red angles are corresponding respect to one of the white transversal lines and the two parallel red lines. Similarly, the two drawn white angles are corresponding.

Moreover, a white angle and a red angle from the same level are corresponding angles with respect to a red transversal and the two white lines. This means that the white angles and the red ones all measure the same. Can you see why?

Finally, the three yellow angles measure the same because they are lower left side angles with respect to the intersection of three lines. Can you tell which ones?

If you have any trouble solving these challenging problems, we invite you to reread this article from the beginning, try to understand each concept very carefully, and come back to these problems later. Be brave and practice a lot!