Parallel Lines Geometry: Finding Angle α Given 120° Measurement

Alternate Angles with Auxiliary Line Construction

Lines a and b are parallel.

What is the size of angle α \alpha ?

aaabbb120α

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:05 First, we'll calculate the angle. Let's get started.
00:08 We know the lines are parallel, as stated in the problem.
00:13 Now, imagine drawing another line that is parallel to these lines.
00:21 Remember, corresponding angles are equal when the lines are parallel.
00:31 To find the angle segment, subtract the right angle from the total.
00:38 Also, alternate angles between parallel lines are equal.
00:43 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Lines a and b are parallel.

What is the size of angle α \alpha ?

aaabbb120α

2

Step-by-step solution

First, let's draw another line parallel to the existing lines that will divide the given angle of 120 degrees in the following way:

aaabbb120α

Note that the line we drew creates two adjacent and straight angles, each equal to 90 degrees.

Now we can calculate the missing part of the angle known to us using the formula:

12090=30 120-90=30

Let's write down the known data as follows:

aaabbbα30

Note that from the drawing we can see that angle alpha and the angle equal to 30 degrees are alternate angles, therefore they are equal to each other.

α=30 \alpha=30

3

Final Answer

30

Key Points to Remember

Essential concepts to master this topic
  • Parallel Lines: When cut by transversal, alternate interior angles are equal
  • Technique: Draw auxiliary line parallel to given lines: 120° - 90° = 30°
  • Check: Verify alternate angles are equal when lines are parallel ✓

Common Mistakes

Avoid these frequent errors
  • Trying to solve without using parallel line properties
    Don't assume α \alpha equals 120° directly = wrong answer! This ignores the geometric relationships. Always identify alternate, corresponding, or co-interior angle pairs when parallel lines are involved.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do we need to draw an auxiliary line?

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The auxiliary line helps us break down the 120° angle into manageable parts. It creates clear angle relationships that we can use with parallel line properties!

What are alternate interior angles?

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Alternate interior angles are on opposite sides of the transversal and inside the parallel lines. They're always equal when lines are parallel.

How do I know which angles are equal?

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Look for angles that are in Z-pattern positions (alternate) or F-pattern positions (corresponding). These are equal when lines are parallel.

Could α \alpha be 60° instead of 30°?

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No! The diagram shows α \alpha is the smaller angle at the bottom intersection. If it were 60°, the geometry wouldn't match the parallel line relationships.

What if I don't see the auxiliary line method?

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You can also use co-interior angles (same-side interior). They add up to 180° when lines are parallel, so α+150°=180° \alpha + 150° = 180° gives α=30° \alpha = 30° .

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