Angles In Parallel Lines

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Parallel and perpendicular lines
Angles in Parallel Lines
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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

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Test yourself on angles in parallel lines!

einstein

In which of the figures are the angles \( \alpha,\beta \) opposite angles?

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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

In which of the figures are the angles \( \alpha,\beta \) opposite angles?

αααβββαααβββαααβββ||||||

Question 2

Which pair of angles is described in the drawing?

aaabbb

Question 3

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

aaabbb

Adjacent Angles

The sum of adjacent angles located between parallel lines is equal to 180180.
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

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

aaabbb

Question 2

Which angles are marked in the figure?

Question 3

\( a \) is parallel to

\( b \)

Determine which of the statements is correct.

αααβββγγγδδδaaabbb

Collateral angles

The sum of consecutive angles located between parallel lines is equal to 180180. 
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:

examples of types of angles
  • Alternate angles
  • Corresponding angles
  • Vertically opposite angles
  • Adjacent angles
  • Consecutive interior angles
  • Consecutive exterior angles

Solution:

  • Examples of alternate angles 1,81, 8
    Both are on different sides and levels, therefore, they are alternate.
  • Examples of corresponding angles 8,48,4
    Both are on the same side and at the same level or floor, therefore, they are corresponding.
  • Examples of vertically opposite angles 1,41,4
    Both share the same vertex and are located opposite each other, therefore, they are vertically opposite angles.
  • Examples of adjacent angles 7,87,8
    Both are on the same line and are located next to each other, therefore, they are adjacent.
  • Examples of exterior alternate angles 1,71,7
    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 3,53,5
    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?

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?

2- 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?

4 -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

examples.example_title

In which of the diagrams are the angles α,β  \alpha,\beta\text{ } vertically opposite?

examples.explanation_title

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.

examples.solution_title

αααβββ

examples.example_title

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

examples.explanation_title

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.

examples.solution_title

No

examples.example_title


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

examples.explanation_title

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.

examples.solution_title

A - corresponding

B - adjacent

examples.example_title

Lines a and b are parallel.

Which of the following angles are co-interior?

αααβββγγγδδδaaabbb

examples.explanation_title

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.

examples.solution_title

β,γ \beta,\gamma

examples.example_title

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

α1α1α1β1β1β1α2α2α2β2β2β2aaabbb

examples.explanation_title

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

examples.solution_title

γ1,γ2 \gamma1,\gamma2

Check your understanding
Question 1

Lines a and b are parallel.

Which of the following angles are co-interior?

αααβββγγγδδδaaabbb

Question 2

Look at the angles shown in the figure below.

What is their relationship?

\( \)αβ

Question 3

The lines in the figure are parallel.

Are the angles \( \alpha \) and \( \beta \) corresponding?

αααβββ

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