Vertically Opposite Angles - Examples, Exercises and Solutions

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

Suggested Topics to Practice in Advance

  1. Perpendicular Lines
  2. Parallel lines

Practice Vertically Opposite Angles

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

examples with solutions for vertically opposite angles

Exercise #1


Look at the rectangle ABCD below.

What type of angles are labeled with the letter A in the diagram?

What type are marked labeled B?

AAABBBCCCDDDBBAA

Step-by-Step Solution

Let's remember the definition of corresponding angles:

Corresponding angles are angles located on the same side of the line that cuts through the two parallels and are also situated at the same level with respect to the parallel line to which they are adjacent.

It seems that according to this definition these are the angles marked with the letter A.

Let's remember the definition of adjacent angles:

Adjacent angles are angles whose formation is possible in a situation where there are two lines that cross each other.

These angles are formed at the point where the intersection occurs, one next to the other, and hence their name.

Adjacent angles always complement each other to one hundred eighty degrees, that is, their sum is 180 degrees.

It seems that according to this definition these are the angles marked with the letter B.

Answer

A - corresponding

B - adjacent

Exercise #2

Look at the rhombus in the figure.

What is the relationship between the marked angles?

BAAB

Step-by-Step Solution

Let's remember the different definitions of angles:

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 they are adjacent to.

Therefore, according to this definition, these are the angles marked with the letter A

Alternate angles are angles located on two different sides of the line that intersects two parallels, and which are also not at the same level with respect to the parallel they are adjacent to.

Therefore, according to this definition, these are the angles marked with the letter B

Answer

A - corresponding; B - alternate

Exercise #3

What angles are described in the drawing?

Step-by-Step Solution

Since the angles are not on parallel lines, none of the answers are correct.

Answer

Ninguna de las respuestas

Exercise #4

Given the parallelogram.

What are alternate angles?

αααγγγδδδβββxxx

Step-by-Step Solution

To solve the question, first we must remember that the property of a parallelogram is that it has two pairs of opposite sides that are parallel and equal.

That is, the top line is parallel to the bottom one.

From this, it is easy to identify that angle X is actually an alternate angle of angle δ, since both are on different sides of parallel straight lines.

Answer

δ,χ \delta,\chi

Exercise #5

Calculate the expression

α+B \alpha+B B30150

Video Solution

Step-by-Step Solution

According to the definition of alternate angles:

Alternate angles are angles located on two different sides of the line that intersects two parallels, and that are also not on the same level with respect to the parallel to which they are adjacent.

It can be said that:

α=30 \alpha=30

β=150 \beta=150

And therefore:

30+150=180 30+150=180

Answer

180 180

examples with solutions for vertically opposite angles

Exercise #1

a is parallel to b.

Calculate the angles shown in the diagram.

115115115111222333444555666777aaabbb

Video Solution

Step-by-Step Solution

Given that according to the definition, the vertex angles are equal to each other, it can be argued that:

115=2 115=2 Now we can calculate the second pair of vertex angles in the same circle:

1=3 1=3

Since the sum of a plane angle is 180 degrees, angle 1 and angle 3 are complementary to 180 degrees and equal to 65 degrees.

We now notice that between the parallel lines there are corresponding and equal angles, and they are:

115=4 115=4

Since angle 4 is opposite to angle 6, it is equal to it and also equal to 65 degrees.

Another pair of alternate angles are angle 1 and angle 5.

We have proven that:1=3=65 1=3=65

Therefore, angle 5 is also equal to 65 degrees.

Since angle 7 is opposite to angle 5, it is equal to it and also equal to 115 degrees.

That is:

115=2=4=6 115=2=4=6

65=1=3=5=7 65=1=3=5=7

Answer

1, 3 , 5, 7 = 65°; 2, 4 , 6 = 115°

Exercise #2

Which angles in the drawing are co-interior given that a is parallel to b?

α1α1α1β1β1β1α2α2α2β2β2β2aaabbb

Video Solution

Step-by-Step Solution

Given that line a is parallel to line b, the anglesα2,β1 \alpha_2,\beta_1 are equal according to the definition of corresponding angles.

Also, the anglesα1,γ1 \alpha_1,\gamma_1 are equal according to the definition of corresponding angles.

Now let's remember the definition of collateral angles:

Collateral angles are actually a pair of angles that can be found on the same side of a line when it crosses a pair of parallel lines.

These angles are on opposite levels with respect to the parallel line they belong to.

The sum of a pair of angles on one side is one hundred eighty degrees.

Therefore, since line a is parallel to line b and according to the previous definition: the angles

γ1​+γ2​=180

are the collateral angles

Answer

γ1,γ2 \gamma1,\gamma2

Exercise #3

According to the drawing

What is the size of the angle? α \alpha ?

120

Video Solution

Step-by-Step Solution

Given that the angle
α \alpha is a corresponding angle to the angle 120 and is also equal to it, thereforeα=120 \alpha=120

Answer

120 120

Exercise #4

Lines a and b are parallel.

Which of the following angles are co-interior?

αααβββγγγδδδaaabbb

Video Solution

Step-by-Step Solution

Let's remember the definition of consecutive angles:

Consecutive angles are, in fact, a pair of angles that can be found on the same side of a straight line when this line crosses a pair of parallel straight lines.

These angles are on opposite levels with respect to the parallel line to which they belong.

The sum of a pair of angles on one side is one hundred eighty degrees.

Therefore, since line a is parallel to line b and according to the previous definition: the anglesβ+γ=180 \beta+\gamma=180

are consecutive.

Answer

β,γ \beta,\gamma

Exercise #5

Calculates the size of the angle α \alpha

α40

Video Solution

Step-by-Step Solution

Let's review the definition of alternate angles between parallel lines:

Alternate angles are angles located on two different sides of the line that intersects two parallels, and that are also not at the same level with respect to the parallel they are adjacent to. Alternate angles have the same value as each other.

Therefore:

α=40 \alpha=40

Answer

40 40

Topics learned in later sections

  1. Angles In Parallel Lines
  2. Alternate angles
  3. Corresponding angles
  4. Collateral angles
  5. Adjacent angles