Math Problem Statement

Given m || n, find the value of x and y. Angles involved: (9x + 2)°, (5x - 18)°, and (2y + 20)°.

Solution

We are given two parallel lines, mnm \parallel n, and asked to find the values of xx and yy using the angles provided in the image. Let’s break down the problem:

Step 1: Identify the relationships between angles

Since mnm \parallel n and the transversal intersects these lines, the following angle relationships apply:

  1. Corresponding angles: Angles on the same side of the transversal and in corresponding positions are equal.
  2. Alternate interior angles: Angles on opposite sides of the transversal but inside the parallel lines are equal.

Step 2: Set up equations

From the image:

  1. The angles (9x+2)(9x + 2)^\circ and (2y+20)(2y + 20)^\circ are corresponding angles, so: 9x+2=2y+209x + 2 = 2y + 20

  2. The angles (9x+2)(9x + 2)^\circ and (5x18)(5x - 18)^\circ are alternate interior angles, so: 9x+2=5x189x + 2 = 5x - 18

Step 3: Solve for xx

We first solve the equation involving xx: 9x+2=5x189x + 2 = 5x - 18 Subtract 5x5x from both sides: 4x+2=184x + 2 = -18 Subtract 2 from both sides: 4x=204x = -20 Divide by 4: x=5x = -5

Step 4: Solve for yy

Substitute x=5x = -5 into the first equation: 9(5)+2=2y+209(-5) + 2 = 2y + 20 Simplify: 45+2=2y+20-45 + 2 = 2y + 20 43=2y+20-43 = 2y + 20 Subtract 20 from both sides: 63=2y-63 = 2y Divide by 2: y=31.5y = -31.5

Final answer:

  • x=5x = -5
  • y=31.5y = -31.5

Would you like further details on any part of this solution?

Here are 5 questions to deepen your understanding:

  1. How do corresponding angles help in solving parallel line problems?
  2. What other angle properties are commonly used in problems involving parallel lines?
  3. Could we have used another pair of angles to solve for xx and yy?
  4. How would the solution change if the lines were not parallel?
  5. Why are alternate interior angles always equal when lines are parallel?

Tip:

Always double-check the relationships between angles in diagrams involving parallel lines, as they frequently provide the key to solving such problems.

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Math Problem Analysis

Mathematical Concepts

Algebra
Angle Relationships
Parallel Lines
Transversal

Formulas

Corresponding angles are equal when two parallel lines are cut by a transversal.
Alternate interior angles are equal when two parallel lines are cut by a transversal.

Theorems

Corresponding Angle Theorem
Alternate Interior Angle Theorem

Suitable Grade Level

Grades 9-10