# Angles In Parallel Lines

🏆Practice 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

## Test yourself on angles in parallel lines!

If one of two corresponding angles is a right angle, then the other angle will also be a right angle.

## 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.

Join Over 30,000 Students Excelling in Math!
Endless Practice, Expert Guidance - Elevate Your Math Skills Today

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:

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

Do you know what the answer is?

## 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:

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
• 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

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.

Alternate

### Exercise #2

$a$ is parallel to

$b$

Determine which of the statements is correct.

### 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.

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

### Exercise #3

Is it possible to have two adjacent angles, one of which is obtuse and the other right?

### 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.

No

### Exercise #4

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.

### Exercise #5

The lines a and b are parallel.

What are the corresponding angles?

### 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 $\alpha=\beta$and therefore the corresponding angles

$\alpha,\beta$