Parallel Lines & Angles: A 90° Intersection Problem

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Let's dive into a geometry problem where we explore angles formed by parallel lines and a transversal! We'll break down a question about Matilde's drawing and figure out how to find the unknown angle. This is super useful for understanding spatial relationships and geometry concepts.

Understanding the Problem

Parallel lines are lines that never intersect, maintaining a constant distance from each other. Think of railroad tracks! A transversal is a line that intersects two or more parallel lines. When a transversal intersects parallel lines, it creates several angles, and these angles have special relationships.

The problem states that Matilde drew two parallel avenues. Imagine them as two perfectly straight roads running side by side. Then, she drew a street that cuts across both avenues. This street is our transversal.

We're given that at the intersection of the street with one of the avenues, the angle formed is 90°. This means the street is perpendicular to that avenue, forming a right angle. The question asks us to find the angle formed at the intersection of the street with the other avenue.

Key Concepts: Parallel Lines and Transversals

When a transversal intersects parallel lines, several pairs of angles are formed, each with specific relationships:

  • Corresponding Angles: These angles are in the same position at each intersection. Corresponding angles are equal.
  • Alternate Interior Angles: These angles are on opposite sides of the transversal and between the parallel lines. Alternate interior angles are equal.
  • Alternate Exterior Angles: These angles are on opposite sides of the transversal and outside the parallel lines. Alternate exterior angles are equal.
  • Same-Side Interior Angles: These angles are on the same side of the transversal and between the parallel lines. Same-side interior angles are supplementary, meaning they add up to 180°.

These relationships are crucial for solving problems involving parallel lines and transversals.

Solving the Problem

In this specific problem, since one angle is 90°, and the avenues are parallel, the street must also be perpendicular to the other avenue. Here's why:

If the first angle is 90°, it means the street is perpendicular to the first avenue. Because the avenues are parallel, any line perpendicular to one must also be perpendicular to the other. Therefore, the angle formed at the intersection of the street with the other avenue must also be 90°.

Consider same-side interior angles. If one angle is 90°, the other angle must be supplementary to it (add up to 180°). So, 180° - 90° = 90°.

Therefore, the correct answer is (B) 90°.

Why the Other Options are Incorrect

Let's look at why the other options don't work:

  • (A) 50°: This is incorrect because if one angle is 90°, and the lines are parallel, the other angle cannot be 50°. The angles must be either equal (corresponding or alternate angles) or supplementary (same-side interior angles).
  • (C) 130°: This is incorrect for the same reason as above. If one angle is 90°, the other angle cannot be 130° due to the properties of parallel lines and transversals.
  • (D) 180°: An angle of 180° would mean the street and avenue form a straight line, which isn't the case in an intersection.

Real-World Applications

Understanding angles formed by parallel lines and transversals isn't just about solving geometry problems. It has real-world applications in various fields:

  • Architecture: Architects use these principles to design buildings and ensure structural integrity. Knowing the angles at which beams and walls meet is crucial for stability.
  • Construction: Construction workers use these concepts when building roads, bridges, and other structures to ensure proper alignment and stability.
  • Navigation: Navigators use angles and parallel lines to determine direction and plot courses. Understanding how angles relate to each other is essential for accurate navigation.
  • Engineering: Engineers apply these principles in various designs, from designing bridges to creating electrical circuits.

Tips for Solving Geometry Problems

Here are some tips to help you solve geometry problems involving parallel lines and transversals:

  • Draw a Diagram: Always start by drawing a clear diagram of the problem. This will help you visualize the relationships between the lines and angles.
  • Identify Parallel Lines and Transversals: Clearly identify the parallel lines and the transversal in the diagram.
  • Apply Angle Relationships: Use the relationships between corresponding angles, alternate interior angles, alternate exterior angles, and same-side interior angles to find unknown angles.
  • Look for Right Angles: If you see a right angle (90°), it can often simplify the problem and help you find other angles.
  • Check Your Work: After solving the problem, check your work to make sure your answer makes sense in the context of the problem.

Conclusion

Understanding the relationships between angles formed by parallel lines and transversals is a fundamental concept in geometry. By applying these principles, you can solve various problems and gain a deeper understanding of spatial relationships. Remember to draw diagrams, identify key elements, and apply the angle relationships to find unknown angles. With practice, you'll become a pro at solving geometry problems!

Now, go forth and conquer those angle problems! You've got this!