Parallel Lines Analysis: Solving the 3α=x Angle Relationship

Q: Why do we need to substitute x = 3α into both angles?

We substitute to see if the alternate angles are equal. If lines are parallel, alternate angles must be equal. By substituting, we can compare $ 4\alpha + 31 $ with $ 4\alpha + 29 $ directly.

Q: What does it mean when we get 31 = 29?

This is a contradiction! Since 31 ≠ 29, the angles are not equal. This proves the lines cannot be parallel because alternate angles between parallel lines must always be equal.

Q: How do I know which angles are alternate angles?

Alternate angles are on opposite sides of the transversal (the crossing line) and between the two lines being tested. In this diagram, the marked angles $ \alpha $ and $ x $ are in alternate positions.

Q: What if I got a different result when substituting?

Double-check your substitution! Remember: if $ x = 3\alpha $, then the first angle becomes $ x + \alpha + 31 = 3\alpha + \alpha + 31 = 4\alpha + 31 $. The second angle stays $ 4\alpha + 29 $.

Parallel Lines with Alternate Angle Verification

Given: $3\alpha=x$

Are they parallel lines?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Are the lines parallel?

00:11 Let's substitute the value of X according to the given data

00:15 Let's solve and substitute in the angle

00:22 Alternate angles are equal between parallel lines

00:26 Let's put the angle values in the equation and solve

00:38 We got an illogical equation

00:41 Therefore the lines are not parallel

00:44 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given: $3\alpha=x$

Are they parallel lines?

Step-by-step solution

If the lines are parallel, the two angles will be equal to each other, since alternate angles between parallel lines are equal to each other.

We will check if the angles are equal by substituting the value of X:

$x+\alpha+31=3\alpha+\alpha+31=4\alpha+31$

Now we will compare the angles:

$4\alpha+31=4\alpha+29$

We will reduce on both sides to $4\alpha$ We obtain:
$31=29$

Since this theorem is not true, the angles are not equal and, therefore, the lines are not parallel.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Alternate angles between parallel lines are always equal
Technique: Substitute $x = 3\alpha$ into both angle expressions and compare
Check: If $4\alpha + 31 = 4\alpha + 29$ then 31 = 29 contradiction ✓

Common Mistakes

Avoid these frequent errors

Assuming lines are parallel without verification
Don't conclude lines are parallel just because you have angle relationships given = wrong assumption! The equal angle condition might not be satisfied. Always substitute the given relationship and check if the resulting angles are actually equal.

Practice Quiz

Test your knowledge with interactive questions

The lines below are not the same size, but are they parallel?

FAQ

Everything you need to know about this question

Why do we need to substitute x = 3α into both angles?

We substitute to see if the alternate angles are equal. If lines are parallel, alternate angles must be equal. By substituting, we can compare $4\alpha + 31$ with $4\alpha + 29$ directly.

What does it mean when we get 31 = 29?

This is a contradiction! Since 31 ≠ 29, the angles are not equal. This proves the lines cannot be parallel because alternate angles between parallel lines must always be equal.

Could the lines still be parallel with different angle relationships?

No - if we're told these are alternate angles and they're not equal, the lines are definitely not parallel. Parallel lines have a very specific property that alternate angles must be equal.

How do I know which angles are alternate angles?

Alternate angles are on opposite sides of the transversal (the crossing line) and between the two lines being tested. In this diagram, the marked angles $\alpha$ and $x$ are in alternate positions.

What if I got a different result when substituting?

Double-check your substitution! Remember: if $x = 3\alpha$ , then the first angle becomes $x + \alpha + 31 = 3\alpha + \alpha + 31 = 4\alpha + 31$ . The second angle stays $4\alpha + 29$ .

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