Find Angle α in Intersecting Lines with β = 65°: Geometry Problem

Q: How do I know if two angles are vertical angles?

Vertical angles are directly opposite each other when two lines intersect. They don't share a side and are separated by the intersection point. Think of them as being across from each other.

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

Vertical angles are opposite and equal. Adjacent angles share a side and add up to 180°. In this problem, α and β are vertical (opposite), not adjacent!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:04 Let's find angle A.

00:10 Remember, vertex angles are equal.

00:14 Plug in the given angle value, and solve for A.

00:19 And that's our solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Find the size of the angle $\alpha$ .

2

Step-by-step solution

To solve this problem, let's identify the angle relationships based on the given information:

Step 1: From the diagram, angle $\beta$ is stated to be $65^\circ$ . It is important to recognize that $\alpha$ and $\beta$ are positioned such that they form a pair of vertical angles.

Step 2: According to the vertical angle theorem, vertical angles are congruent. This means that if $\beta = 65^\circ$ , then $\alpha$ must also equal $65^\circ$ because they are vertical angles created by intersecting lines.

Therefore, the size of angle $\alpha$ is $\mathbf{65^\circ}$ .

3

Final Answer

65

Key Points to Remember

Essential concepts to master this topic

Vertical Angle Theorem: Opposite angles formed by intersecting lines are equal
Technique: Identify vertical angle pairs - if β = 65°, then α = 65°
Check: Verify angles are across from each other at intersection point ✓

Common Mistakes

Avoid these frequent errors

Confusing adjacent angles with vertical angles
Don't assume α + β = 180° just because they look close = wrong answer of 115°! Adjacent angles are supplementary, but vertical angles are equal. Always identify if angles are opposite each other across the intersection point.

Practice Quiz

Test your knowledge with interactive questions

Can a triangle have a right angle?

FAQ

Everything you need to know about this question

How do I know if two angles are vertical angles?

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Vertical angles are directly opposite each other when two lines intersect. They don't share a side and are separated by the intersection point. Think of them as being across from each other.

Why are vertical angles always equal?

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When two lines intersect, they create four angles. Each pair of opposite angles must be equal because they're formed by the same two intersecting lines. It's a fundamental property of geometry!

What's the difference between vertical and adjacent angles?

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Vertical angles are opposite and equal. Adjacent angles share a side and add up to 180°. In this problem, α and β are vertical (opposite), not adjacent!

Do I need to do any calculations for vertical angles?

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Usually not! Once you identify that two angles are vertical, they're automatically equal. If one angle is 65°, its vertical angle is also 65°. No arithmetic needed!

What if the problem gives me an expression instead of a number?

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Set the expressions equal to each other! If one vertical angle is $3x + 10$ and the other is $5x - 20$ , then $3x + 10 = 5x - 20$ . Solve for x first.

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Find Angle α in Intersecting Lines with β = 65°: Geometry Problem

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