Corresponding angles

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 $A$ and $B$ with a transversal $C$ cutting both of them.

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".

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".

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 $180^o$ 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".

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

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.

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

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

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

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

Solution:

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

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

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

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

Then:

Angle D measures $45°$.

Angle C measures $105°$.

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

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

Solution:

After briefly observing the image we will see that, the segment $AB$ is parallel not only to the edge $NK$, but also to the edge (or side) $LM$. 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 $LM$ is also parallel to the edge $AB$.

Now we will find in the image the angle corresponding to the angle $L$. Looking quickly we can say that the angle corresponding to the angle $L$ is the $KAB$. 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 ($AB$ and $LM$) and the angles are also at the same level with respect to the parallel line.

Answer:

The angle corresponding to angle $L$ is $KAB$.

Given the isosceles triangle ABC

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

Question:

Is it possible to check that the triangle $\triangle AED$ 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 $\triangle ABC$ and $\triangle ACE$ are equal (because they face equal sides), they are supplementary and equal to the angles $\sphericalangle AED$ and $\sphericalangle ADE$.

Answer:

Therefore, the triangle $\triangle AED$ is isosceles.

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:

$3X-10=2X+30$

$3X-2X=30+10$

$X=40$

Thus we find the value of $X$.

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

And which ones are marked with the letter $Y$?

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

Solution:

Identification and definition of elements.

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

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

And which ones with the letter $ABCD$?

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

Answer:

Corresponding / adjacent

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

Task:

Find the value of the angle $\alpha$

Solution:

First we identify the angle $53^o$, 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 $a$ and $b$ is equal to $180^o$, so then we would have the following equation:

$\alpha + 78º + 53º = 180º$

Subsequently subtracting the angle, we have the following:

$\alpha = 180º - 78º - 53º$

$\alpha = 49º$

Answer:

$\alpha=49º$

Given the polygon in the figure

Task:

Which of the straight pairs is parallel to each other?

Solution:

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

$30°+150°=180°$

• 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.

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

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

Assignment:

Find the value of $α$

Solution:

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

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

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

$α=β-δ$

$α=130°-45°$

$α=85°$

Answer:

$α=85°$

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:

• Angle notation
• Sides, vertices, and angles
• Right angles
• Acute angle
• Obtuse angle
• Plane angles
• Adjacent angles
• Angles opposite at the vertex
• Sum of the angles of a triangle
• Sum of the angles of a polygon
• Sum and difference of angles

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