Corresponding angles

🏆Practice angles in parallel lines

The corresponding angles are those that are on the same side of the transversal that cuts two parallel lines and are at the same level with respect to the parallel line. The corresponding angles are of the same size.

The following image illustrates two pairs of corresponding angles, the first ones have been painted red and the others blue.

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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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What are the corresponding angles?

Before offering the specific explanation about the corresponding angles it is necessary to understand in which cases these angles can be formed. The basic way to describe it is with a diagram of two parallel lines with a transversal that cuts them (if you need more details it is convenient to consult the specific article that deals with the topic of "Parallel lines"), as can be seen in this illustration:

As mentioned, there are two parallel lines A A and B B with a transversal C C cutting both of them.


Other types of angles

There are other types of angles that are formed in cases like the one we have just discussed. We will analyze them briefly:

Alternate angles

These are the angles that are on opposite sides of the transversal that cuts two parallel lines and are not on the same side with respect to the parallel line. Alternate angles are the same size.

For more details go to the specific article that deals with the subject of "alternate angles".

1st Image Alternate angles


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Angles opposite at the vertex

They are formed by two straight lines that intersect, have a vertex in common and are opposite each other. The angles opposite at the vertex are of the same size.

For more details go to the specific article that deals with the topic of "angles opposite at the vertex".

angles opposite the vertex


Collateral angles

They are the angles that are on the same side of the transversal that cuts two parallel lines and are not at the same level with respect to the parallel line. Together they complete 180o 180^o degrees, that is, the sum of two collateral angles is equal to one hundred and eighty degrees.

For more details go to the specific article that deals with the subject of "collateral angles".



Do you know what the answer is?

Exercises with corresponding angles

Exercise 1

In each of the following illustrations indicate if they are corresponding angles or not. In both cases explain why.

illustration 1 new

illustration 2 new

illustration 3 new

Solution:

Diagram No 1:

In this case we are really dealing with corresponding angles since they meet the two criteria of their definition, i.e., they are two angles that are on the same side of the transversal that cuts the two parallel lines and the angles are on the same side with respect to the parallel line.

Diagram No. 2:

In this case we are not dealing with corresponding angles since they do not meet the criteria of their definition, i.e., we are dealing with two angles that are on both sides of the transversal that cuts the two parallel straight lines and the two angles are not on the same side with respect to the parallel straight line.

Diagram No. 3:

In this case we are really dealing with corresponding angles since they meet the two criteria of their definition, i.e., we are dealing with two angles that are on the same side of the transversal that cuts the two parallel lines and the angles are on the same side with respect to the parallel line.
Then:

Scheme No 1: corresponding angles

Scheme No 2: they are not corresponding angles, however, they are internal alternate angles.

Scheme No 3: corresponding angles.


Exercise 2

Given the triangle BCD \triangle BCD as illustrated in the following image:.

Corresponding_angles_-_Exercise_02.original

The angle B B of the triangle BCD \triangle BCD is equal to 30o 30^o .

Also, it is known that, the line KL KL inside the triangle is parallel to the edge (or side) of the triangle and the angle K of the triangle BLK is equal to 45o 45^o .

Find the other two angles of the triangle BCD \triangle BCD .

Solution:

Looking at the picture we see that, we have two parallel lines (KL KL and DC DC ) which are cut by a transversal (the edge DB DB ). The angle D D of the triangle is equal to the angle BKL BKL since they are corresponding angles, that is to say, they are two angles located on the same side of the transversal (DB DB ) that cuts the two parallel lines (KL KL and DC DC ) and these angles are on the same side with respect to the parallel line.

From the above we deduce that the angle D D of the triangle is equal to 45° 45° .

We also know that the sum of the three angles of any triangle equals 180° 180° .

Therefore, angle C equals 180°30°45°=105° 180°-30°-45°=105° .

Then:

Angle D measures 45° 45° .

Angle C measures 105° 105° .


Check your understanding

Exercise 3

Given the parallelogram KLMN KLMN . Also, we know that the segment AB AB is parallel to the edge NK NK .

Corresponding_angles_-_Exercise_03.original

Find the angle corresponding to the angle L L , highlighted in the diagram.

Solution:

After briefly observing the image we will see that, the segment AB AB is parallel not only to the edge NK NK , but also to the edge (or side) LM LM . The idea here is that these are two opposite edges of the parallelogram that have the same length and are parallel to each other. Therefore, the edge LM LM is also parallel to the edge AB AB .

Now we will find in the image the angle corresponding to the angle L L . Looking quickly we can say that the angle corresponding to the angle L L is the KAB KAB . As we know that, this angle together with angle L meet the two criteria of the definition of the corresponding angles, i.e., they are two angles located on the same side of the transversal (edge KL) that cuts the two parallel lines (AB AB and LM LM ) and the angles are also at the same level with respect to the parallel line.

Answer:

Corresponding_angles_-_Exercise_03_-_Soluc.original

The angle corresponding to angle L L is KAB KAB .


Exercise 4

Given the isosceles triangle ABC

Inside it, in the figure there is a line ED ED which is parallel of CB CB .

Exercise 2 Given an isosceles triangle ABC

Question:

Is it possible to check that the triangle AED \triangle AED is also isosceles?
Solution :

To check that the triangle is isosceles, it is necessary to check that the sides are equal or that the opposite angles are equal.

Since the triangles ABC \triangle ABC and ACE \triangle ACE are equal (because they face equal sides), they are supplementary and equal to the angles AED \sphericalangle AED and ADE \sphericalangle ADE .

Answer:

Therefore, the triangle AED \triangle AED is isosceles.


Do you think you will be able to solve it?

Exercise 5

What is the value of X X ?

Exercise 5 What is the value of X

Solution:

The given angles are corresponding angles, so they are equal.

That is, all that is needed is to solve the following resulting equation:

3X10=2X+30 3X-10=2X+30

3X2X=30+10 3X-2X=30+10

X=40 X=40

Thus we find the value of X X .


Exercise 6

Answer the question knowing that ABCD is a rectangle.

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

And which ones are marked with the letter Y Y ?

Answer the question knowing that ABCD ABCD is a rectangle.

Solution:

Identification and definition of elements.

Since we have to answer the question knowing that ABCD ABCD is a rectangle.

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

And which ones with the letter ABCD ABCD ?

  1. Corresponding / adjacent
  2. Complementary / alternate
  3. Supplementary / adjacent
  4. Opposite by vertex / Opposite by vertex.

Answer:

Corresponding / adjacent


Test your knowledge

Exercise 7

Given that a a , b b and c c are parallel

Exercise 7 Given that a,b and c are parallel

Task:

Find the value of the angle α \alpha

Solution:

First we identify the angle 53o 53^o , using the property of angle opposite by vertex we write that value in the opposite part of the angle.

On the other hand, we know that the sum of the internal angles of the triangle formed by the transversal lines that cut the parallel lines a a and b b is equal to 180o 180^o , so then we would have the following equation:

α+78º+53º=180º \alpha + 78º + 53º = 180º

Subsequently subtracting the angle, we have the following:

α=180º78º53º \alpha = 180º - 78º - 53º

α=49º \alpha = 49º

Answer:

α=49º \alpha=49º


Exercise 8

Given the polygon in the figure

Exercise 7 Given the polygon of the figure

Task:

Which of the straight pairs is parallel to each other?

Solution:

  • Between a a and b b passes a line summing alternating angles whose equality can be checked.

30°+150°=180° 30°+150°=180°

  • Between b and g we can identify an internal alternate angle that is not equal, so it is not a parallel line.
  • Between b and d there are corresponding angles that are not equal, therefore, they are not parallel lines.
  • There is no data on e and b because they are not crossed by a straight line.

Do you know what the answer is?

Exercise 9

The following figure shows three parallel lines a a , b b and c c .

Given that a a , b b and c c are parallel.

a,b and c are parallel

Assignment:

Find the value of α α

Solution:

We assign with the letter β β to the angle with which we have a correspondence with the angle 130o 130^o , as shown in the picture.

The angle β β and the angle 130o 130^o are corresponding and are therefore equal.

The angle δ δ and the angle 45o 45^o are internal alternating angles and therefore are equal and you have the following equality:

α=βδ α=β-δ

α=130°45° α=130°-45°

α=85° α=85°

Angle β and angle 130

Answer:

α=85° α=85°


Questions on the subject:

What does corresponding angles mean?

They are the ones that are on the same side of the transversal that cuts two parallel lines and are on the same side with respect to the parallel line.


What is the combination of the corresponding angles?

Their value is the same because they are on the same side with respect to the same parallel lines.


What do the corresponding sides or angles in the triangles mean?

The corresponding angles will have the same measure in congruent triangles.


How long are the corresponding angles?

They measure the same.


What are the corresponding angles and what are their characteristics?

They are non-adjacent angles located on the same side of the transversal line that cuts the parallels and their main characteristic is that they are equal.


What is the corresponding side?

They are those that have the same length in congruent triangles.


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

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


Check your understanding

examples with solutions for corresponding 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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