Vertically Opposite Angles: Relationship Between Acute and Obtuse Pairs

Q: What exactly are vertically opposite angles?

When two lines intersect, they create four angles. Vertically opposite angles are the pairs that are directly across from each other, not next to each other.

Q: Can one vertically opposite angle be acute and the other obtuse?

No, never! Since vertically opposite angles are always equal, they must be the same type. If one is $60^\circ$ (acute), the other is also $60^\circ$ (acute).

Q: What's the difference between vertically opposite and adjacent angles?

Adjacent angles are next to each other and add up to $180^\circ$. Vertically opposite angles are across from each other and are equal. Don't mix them up!

Q: If two lines intersect at 90°, what happens to vertically opposite angles?

All four angles would be $90^\circ$ (right angles)! The vertically opposite angles are still equal - they're just all the same measure in this special case.

Q: How can I remember this property easily?

Think "opposite twins" - vertically opposite angles are like identical twins, they're always exactly the same! This makes the statement in the question impossible.

Vertically Opposite Angles with Equal Measures

If one vertically opposite angle is acute, then the other will be obtuse.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's decide if this statement is true or false.

00:09 Remember, vertical angles are always equal.

00:13 So, if one angle is acute, the other must be acute too.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

If one vertically opposite angle is acute, then the other will be obtuse.

Step-by-step solution

To solve this problem, we need to understand the properties of vertically opposite angles:

Vertically opposite angles are the angles that are opposite each other when two lines intersect.
One key property of vertically opposite angles is that they are always equal in measure.
An acute angle is defined as an angle that is less than $90^\circ$ .
An obtuse angle is defined as an angle that is greater than $90^\circ$ .

Given that vertically opposite angles are equal, if one angle is acute, the opposite angle must also be acute. This contradicts the statement in the problem that if one is acute, the other will be obtuse.

Therefore, the correct analysis of the problem reveals that the statement is incorrect.

Thus, the solution to the problem is False.

Final Answer

False

Key Points to Remember

Essential concepts to master this topic

Rule: Vertically opposite angles are always equal in measure
Technique: If one angle is $30^\circ$ , the opposite angle is also $30^\circ$
Check: Both angles must be same type: acute-acute or obtuse-obtuse ✓

Common Mistakes

Avoid these frequent errors

Assuming opposite angles are supplementary
Don't think vertically opposite angles add up to 180° = wrong angle types! This confuses them with adjacent angles on a line. Always remember vertically opposite angles are equal, so they're the same type (both acute or both obtuse).

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

What exactly are vertically opposite angles?

When two lines intersect, they create four angles. Vertically opposite angles are the pairs that are directly across from each other, not next to each other.

Can one vertically opposite angle be acute and the other obtuse?

No, never! Since vertically opposite angles are always equal, they must be the same type. If one is $60^\circ$ (acute), the other is also $60^\circ$ (acute).

What's the difference between vertically opposite and adjacent angles?

Adjacent angles are next to each other and add up to $180^\circ$ . Vertically opposite angles are across from each other and are equal. Don't mix them up!

If two lines intersect at 90°, what happens to vertically opposite angles?

All four angles would be $90^\circ$ (right angles)! The vertically opposite angles are still equal - they're just all the same measure in this special case.

How can I remember this property easily?

Think "opposite twins" - vertically opposite angles are like identical twins, they're always exactly the same! This makes the statement in the question impossible.

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