Angles In Parallel Lines

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.

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.

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.

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.

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:

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.