Math Problem Statement

In the figure shown, line q is a transversal of parallel lines l, m, n, and p. What are the values of x and y?

Solution

Let's analyze the problem step by step.

We are given a figure where qq is a transversal of the parallel lines ll, mm, nn, and pp. The angles formed by the transversal with these lines are labeled as xx^\circ, 3030^\circ, and yy^\circ.

Understanding the Relationships:

  • Alternate Interior Angles: These angles are equal when a transversal crosses parallel lines.
  • Corresponding Angles: These angles are also equal when a transversal crosses parallel lines.
  • Linear Pair: The angles on a straight line sum up to 180°.

Step 1: Use Alternate Interior Angles

Since lpl \parallel p and qq is the transversal:

  • The angle xx^\circ is alternate to the angle yy^\circ.

So, x=yx = y.

Step 2: Use Linear Pair

Given that the angle between lines mm and nn on the transversal is 3030^\circ, and the adjacent angle is yy^\circ, we can write:

y+30=180y + 30^\circ = 180^\circ

y=150y = 150^\circ

Step 3: Conclude for xx

Since x=yx = y, it follows that:

x=150x = 150^\circ

Final Answer:

The correct values of xx and yy are x=150x = 150^\circ and y=150y = 150^\circ. Therefore, the correct answer is:

Dx=150, y=150\boxed{D} \quad x = 150^\circ, \ y = 150^\circ

Would you like more details on any part of the solution?

Relative Questions:

  1. How do alternate interior angles relate to parallel lines?
  2. What is the significance of corresponding angles in parallel lines?
  3. Can you explain how the linear pair relationship works in this problem?
  4. How would the solution change if yy were given instead of 3030^\circ?
  5. What are some other properties of transversals with parallel lines?

Tip:

When working with parallel lines and a transversal, always look for alternate interior angles and corresponding angles first, as these relationships can simplify the problem significantly.

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Math Problem Analysis

Mathematical Concepts

Geometry
Parallel Lines
Transversals
Alternate Interior Angles
Corresponding Angles
Linear Pair

Formulas

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Theorems

Alternate Interior Angles Theorem

Suitable Grade Level

Grades 7-9