$$ a $$ is parallel to $$ b $$ Determine which of the statements is correct. α α α β β β γ γ γ δ δ δ a a a b b b

$$ \beta,\gamma $$ Colaterales$$ \alpha,\delta $$ Alternates

$$ \alpha,\beta $$ Colaterales$$ \gamma,\delta $$ Adjacent

$$ \alpha,\beta $$ Adjacent $$ \gamma,\delta $$ Alternates

Angles In Parallel Lines

Angles on Parallel Lines

If we add a third line that intersects the two parallel lines (those lines that could never cross), we will obtain various types of angles.
To classify these angles we must observe if they are:
above the line - the pink part
below the line - the light blue part
to the right of the line - the red part
to the left of the line - the green part

Detailed explanation

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Test yourself on angles in parallel lines!

Which type of angle is described in the figure below?:

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Corresponding angles

The corresponding angles located between parallel lines are equal.
They are called corresponding angles because:

they are on the same side of the transversal
they are on the same "floor" in relation to the line

Here are some examples of corresponding angles:

The two angles marked are on the left side and on the ground floor - they are corresponding and equivalent.

Vertically Opposite Angles

The vertically opposite angles that are located between parallel lines are equal.
They are called vertically opposite angles because:

They share the same vertex - they are located on the same vertex
They are opposite each other

Here are some examples of vertically opposite angles:

The two marked angles are located on the same vertex and are opposite each other, therefore, they are vertically opposite angles.

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Test your knowledge

Question 1

What angles are described in the drawing?

Question 2

Which type of angles are shown in the figure below?

Question 3

Which type of angles are shown in the diagram?

Adjacent Angles

The sum of adjacent angles located between parallel lines is equal to $180$ .
They are called adjacent angles because:

They are next to each other
They are on the same line

Here are some examples of adjacent angles:

The two marked angles are on the same line - (diagonal) - and are next to each other, therefore, they are adjacent angles.

Alternate Angles

The alternate angles that are located between parallel lines are equal.
They are called alternate angles because:

they are not on the same side of the transversal
they are not on the same "level" in relation to the line

Here are some examples of alternate angles:

two pairs of alternate angles, one is painted red and the other blue

The two marked angles are on different "levels" and sides, therefore, they are alternate angles.

Do you know what the answer is?

Question 1

Which pair of angles is described in the drawing?

Question 2

What type of angles are shown below given that lines a and b are parallel?

Question 3

What type of angles are shown below given that a and b are parallel lines?

Collateral angles

The sum of consecutive angles located between parallel lines is equal to $180$ .
They are called consecutive angles because:

they are on the same side of the transversal
but they are not on the same "level" in relation to the line

Here are some examples of consecutive angles:

A10 - External and internal collateral angles

The two marked angles are on the same side of the line, but at a different height, therefore, they are consecutive angles.
Observe: The angles painted in red in the illustration above are external consecutive angles since they are on the outer side of the parallel lines
The internal consecutive angles are on the inner side of the parallel lines:

Angles on Parallel Lines

Now to practice!

Give an example according to the illustration of:

Alternate angles
Corresponding angles
Vertically opposite angles
Adjacent angles
Consecutive interior angles
Consecutive exterior angles

Solution:

Examples of alternate angles $1, 8$
Both are on different sides and levels, therefore, they are alternate.
Examples of corresponding angles $8,4$
Both are on the same side and at the same level or floor, therefore, they are corresponding.
Examples of vertically opposite angles $1,4$
Both share the same vertex and are located opposite each other, therefore, they are vertically opposite angles.
Examples of adjacent angles $7,8$
Both are on the same line and are located next to each other, therefore, they are adjacent.
Examples of exterior alternate angles $1,7$
Both are on the same side, but not at the same level. In addition, they are located on the outside of the line, therefore, they are exterior alternate angles.
Examples of interior alternate angles $3,5$
Both are on the same side, but not at the same level. In addition, they are located on the inside of the line, therefore, they are interior alternate angles.

Another exercise:

What are the marked angles called in the illustration?

Solution

The marked angles are alternate
They are located on different sides and heights, therefore, they are alternate.

Another exercise:

What are the angles shown in the illustration called?

Solution

The indicated angles are consecutive
They are on the same side of the line, but at different heights, therefore, they are external consecutive angles.

Another exercise:

What are the angles shown in the illustration called?

Solution

The indicated angles are adjacent
They are on the same blue line and are next to each other, therefore, they are adjacent angles.

Examples and exercises with solutions of right angles and parallels

Exercise #1

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

Exercise #2

In which of the diagrams are the angles $\alpha,\beta\text{ }$ 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

Exercise #3

$a$ is parallel to

$b$

Determine which of the statements is correct.

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

$\beta,\gamma$ Colaterales $\gamma,\delta$ Adjacent

Exercise #4

Which type of angle is described in the figure below?:

Step-by-Step Solution

Let's remember that adjacent angles are angles that are formed when two lines intersect each other.

These angles are created at the point of intersection, one adjacent to the other, and that's where their name comes from.

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

Answer

Adjacent

Exercise #5

What angles are described in the drawing?

Step-by-Step Solution

Let's remember that vertical angles are angles that are formed when two lines intersect, and they are created at the point of intersection, opposite each other.

Answer

Vertices

Check your understanding

Question 1

Which angles are marked in the figure?

Question 2

$$ a $$ is parallel to

$$ b $$

Determine which of the statements is correct.

Question 3

Lines a and b are parallel.

Which of the following angles are co-interior?

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