Parallel Lines Geometry: Find α and β Using Given Angles 104° and 81°
Alternate and Corresponding Angles with Parallel Lines
Determine the value of the α-and-β- angles shown in the below diagram:
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Understand the problem
Determine the value of the α-and-β- angles shown in the below diagram:
Step-by-step solution
In the question, we can observe that there are two pairs of parallel lines, lines a and b.
When a line crosses two parallel lines, different angles are formed
Angles alpha and the given angle of 104 are on different sides of the transversal line, but both are in the interior region between the two parallel lines,
This means they are alternate angles, and alternate angles are equal.
Therefore,
Angle beta and the second given angle of 81 degrees are both on the same side of the transversal line, but each is in a different position relative to the parallel lines, one in the exterior region and one in the interior. Therefore, we can conclude that these are corresponding angles, and corresponding angles are equal.
Therefore,
Final Answer
Key Points to Remember
- Alternate Interior Angles: Equal when formed by parallel lines and transversal
- Corresponding Angles: Equal positions like α = 104° and β = 81°
- Check: Verify angle relationships match parallel line properties ✓
Common Mistakes
- Confusing alternate and corresponding angle positions Don't assume all angles are equal just because lines are parallel = wrong angle values! Different angle relationships (alternate, corresponding, co-interior) have different rules. Always identify the specific angle relationship first, then apply the correct property.
Practice Quiz
Does the drawing show an adjacent angle?
FAQ
How do I tell if angles are alternate interior or corresponding?
+Alternate interior angles are on opposite sides of the transversal and between the parallel lines. Corresponding angles are in the same relative position at each intersection.
Are all angles formed by parallel lines equal?
+No! Only corresponding angles and alternate interior angles are equal. Co-interior angles are supplementary (add up to 180°), not equal.
What if I can't see which lines are parallel in the diagram?
+Look for arrow markers or labels like 'a' and 'b'. Parallel lines are often marked with identical symbols or explicitly stated in the problem.
Why is α = 104° and not some other relationship?
+Because α and the 104° angle are alternate interior angles - they're on opposite sides of the transversal but both between the parallel lines, so they must be equal.
How do I remember the difference between angle types?
+Alternate = opposite sides, Corresponding = same position, Co-interior = same side and add to 180°. Draw the 'Z', 'F', and 'C' patterns to help visualize!
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