Vertically Opposite Angles

🏆Practice angles in parallel lines

What are opposite angles?

Before going deeper into opposite angles, we will pause a moment to visualize the types of scenarios where this type of angle can be found. To make it easier to understand, we will draw two parallel straight lines cut by a secant or transversal, as shown in the following illustration:

A2 - Parallel lines

What do we see here? The transversal C C intersects with each one of the straight lines A A and B B (in our case A A and B B are parallel, although this is not required in order to get opposite angles).

With this example in mind, we are ready to move on to the formal definition of opposite angles, which will help us to identify them more easily:

Opposite angles are a pair of angles that arise when two straight lines intersect. These angles are formed at the point of intersection (which we will call the vertex), one in front of the other. Opposite angles are equal.

In the following illustration, we can see two examples of opposite angles, the first pair is marked in red and the second pair in blue.

C - Opposite angles

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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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More types of angles

Opposite angles are not the only types of angles that we can find in geometric problems. Even the example seen in the introduction actually includes several other types of angles that we will mention briefly below:

Corresponding angles

Corresponding angles are a pair of angles that can be found on the same side of a transversal intersecting two parallel lines. These angles are on the same side of the parallel lines to which they belong. Corresponding angles are equal.

If you wish to read a more extensive explanation, you can consult the article on the subject "Corresponding angles".

A3 - examples of corresponding angles

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Alternate angles

Alternate angles are a pair of angles that can be found on opposite sides of a transversal intersecting two parallel lines. These angles are on the opposite sides of the parallel lines to which they belong. Alternate angles are equal.

Alternate angles

For a more detailed explanation, please refer to the article "Alternate angles".

Collateral angles

Collateral angles are a pair of angles that can be found on the same side of a transversal intersecting two parallel lines. These angles are on the opposite side of the parallel lines to which they belong. The sum of the collateral angles equals 180º 180º .

For a more detailed explanation, please refer to the article "Collateral angles".

A10  - External and internal collateral angles

Do you know what the answer is?

Exercise problems with opposite angles

Exercise 1

In each of the following diagrams, determine if the angles are opposite angles, and if not, specify the type of angle.

Scheme1:

image 1 new

Scheme 2:

image 2 new

Scheme 3:

image 3 new

Solution:

Scheme 1:

In this scheme if we are dealing with opposite angles. The reason is that they meet the criteria, that is to say, two straight lines intersect and the opposite angles are formed at the point of intersection, one in front of the other.

Scheme 2:

In this scheme we are not dealing with opposite angles, but rather with corresponding angles. The reason is that the two angles are on the same side of the transversal that cuts the two parallel lines. Moreover, these angles are on the same side of the parallel lines to which they belong.

Scheme 3:

In this scheme we are not dealing with opposite angles either, but rather collateral angles. The reason is that the two angles are on the same side of the transversal that cuts the two parallel lines. Moreover, these angles are at the opposite side of the parallel lines to which they belong.

So:

Scheme 1: opposite angles

Scheme 2: corresponding angles

Scheme 3: collateral angles


Exercise 2

Given the parallelogram ABCD as illustrated in the schematic diagram

Given the parallelogram ABCD ABCD as illustrated in the diagram.

The point K K is the point at which the diagonals meet in the parallelogram. ABCD ABCD .

The angle AKD AKD measures 30º 30º .

The angle KBC KBC measures 50º 50º .

Based on the given information calculate the angle BCK BCK .

Solution:

First of all, to help us find the answer, we will label the angles as follows:

We will name the angle AKDK1 AKD K1 (measure 30º 30º )

We will name the angle BKCK2 BKC K2

We will name the angle KBCB1 KBC B1 (measures 50º 50º )

We will name the angle BCKC1 BCK C1 (the angle we are looking for)

First, we will focus on the triangle BCK BCK since the angle C1 C1 is in it.

We will build on what we already know. The sum of the interior angles of a triangle is equal to 180º 180º . The angle B1 B1 measures 50º 50º according to the given information. That is, if we can find the measure of the angle K2 K2 then we can calculate angle C1 C1 .

As can be understood from the data and the illustration, the point K K is the intersection of the diagonals AC AC and BD BD in the parallelogram ABCD ABCD . We can see that, according to its definition, this intersection forms the opposite angles K1 K1 and K2 K2 . The opposite angles are equal, therefore k1=k2=30º k1 = k2 = 30º .

Now we can go back to the triangle BCK BCK and find angle C1 C1 :

C1=180ºB1K2=180º50º30º=100º C1 = 180º − B1 − K2 = 180º − 50º − 30º = 100º

That is, the angle C1 C1 which is in fact the angle we are looking for BCK BCK measures 100º 100º .

Then:

The angle BCK BCK measures 100º 100º .


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Exercise 3

Given the isosceles trapezoid ABCD as illustrated in the following diagram

Given the isosceles trapezoid ABCD ABCD as illustrated in the diagram.

The point M M is the point where the diagonals of the trapezoid meet. ABCD ABCD .

MA=MB MA = MB

The angle DMC DMC measures 120º 120º .

Based on the given information, calculate the angles of the triangle. ABM ABM .

Solution:

First of all, to help us find the answer, we will label the angles as follows:

We will name the angle DMCM1 DMC \: M1 (measure 120º 120º )

We will name the angle AMBM2 AMB \: M2 (one of the angles we are looking for)

We will name the angle MABA1 MAB \: A1 (one of the angles we are looking for)

We will name the angle ABMB1 ABM \: B1 (one of the angles we are looking for)

We will focus on triangle ABM since we must find its angles.

We will start with angle M2 M2 .

As we can see from the data and the illustration, the point M M is the intersection of the diagonals AC AC and BD BD in the trapezoid ABCD ABCD . According to its definition, this intersection point forms opposite angles M1 M1 and M2 M2 . Opposite angles are equal, therefore, M1=M2=120º M1= M2 = 120º .

Now we turn to another piece of information we have, the side MA=MB MA = MB . This implies that the triangle ABM ABM is an isosceles triangle. We know that in an isosceles triangle the two base angles are equal.

That is to say,

A1=B1 A1 = B1

To calculate the angles, remember that the sum of the interior angles of a triangle is 180º 180º and that the measure of M2 M2 is already known.

Therefore, that gives us:

180º=A1+B1+M2=2A1+120º 180º = A1 + B1 + M2 = 2*A1 + 120º

A1=(180º120º)/2=30º A1 = (180º - 120º) / 2 = 30º

Then:

The angle AMB AMB (angle M2 M2 ) measures 120º 120º .

The angle MAB MAB (angle A1 A1 ) measures 30º 30º .

The angle ABM ABM (angle B1 B1 ) measures 30º 30º .


Exercise 4

What is the value of X knowing that the lines are parallel?

What is the value of X X given that the lines are parallel?

Solution:

The marked angles are exterior angles to the parallel lines. We can use our knowledge of opposite angles, which share the same properties.

If we take one of the opposite angles and the angle external to the other parallel line, we see that they are actually supplementary angles and therefore are equal.

That is,

X+70º=2X X + 70º = 2X

70º=2XX 70º = 2X - X

70º=X 70º = X

Then:

We find that the value of X=70º X = 70º .


Do you think you will be able to solve it?

Exercise 5

Knowing that CE CE is parallel to AD AD , find the value of X X .

the angle ∡BCE = X being opposite at the vertex with the angle measuring X

Solution:

We have that the angle BCE=X ∡BCE = X as it is opposite to angle X X . The same is true for the angle BAD=X10 ∡BAD = X - 10 .

Then, due to the parallel lines we get: ABC=BAD+BCE ∡ABC = ∡BAD + ∡BCE that is , 3X30=X+(X10) 3X - 30 = X + (X - 10) .

Then solving for the value of X X we get:

X=20º X = 20º

Therefore:

X=20º X = 20º


Review questions:

What are opposite angles?

They are angles that are formed when two straight lines intersect, one in front of the other just at the point of intersection.


What is the main characteristic of opposite angles?

The main characteristic is that opposite angles are equal.


In an illustration of parallel lines cut by a transversal, which pair of angles have the same property of the opposite angles, that is, that they are equal?

The corresponding angles and the alternate angles.


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

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


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examples with solutions for vertically opposite 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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