Identifying Vertically Opposite Angles: Alpha and Beta in Multiple Diagrams

Vertically Opposite Angles with Line Intersections

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

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1

Understand the problem

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

2

Step-by-step solution

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.

3

Final Answer

αααβββ

Key Points to Remember

Essential concepts to master this topic
  • Definition: Vertically opposite angles form when two lines intersect, facing each other
  • Technique: Look for angles across from each other at intersection point
  • Check: Vertically opposite angles are always equal in measure ✓

Common Mistakes

Avoid these frequent errors
  • Confusing adjacent angles with vertically opposite angles
    Don't assume any two angles at an intersection are vertically opposite = wrong identification! Adjacent angles share a side and are supplementary (add to 180°). Always look for angles that are directly across from each other with no shared sides.

Practice Quiz

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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

How can I tell if two angles are vertically opposite?

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Look for angles that are directly across from each other at the intersection point. They should have no shared sides and be separated by the other two angles.

Are vertically opposite angles always equal?

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Yes, always! This is a fundamental property. If α \alpha and β \beta are vertically opposite, then α=β \alpha = \beta .

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

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Adjacent angles share a common side and add up to 180°. Vertically opposite angles are across from each other, don't share sides, and are equal in measure.

Can I have vertically opposite angles without two straight lines?

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No! Vertically opposite angles only form when two straight lines intersect. You need exactly two lines crossing to create four angles, where opposite pairs are vertically opposite.

In the diagrams, why is only the second one correct?

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In diagram B, α \alpha and β \beta are directly across from each other where two straight lines intersect. In the other diagrams, the angles are either adjacent or don't have the proper line intersection.

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