Math Problem Statement

In this diagram, transversal h cuts parallel lines f and g. What is the measure of angle J?

Solution

To solve for the measure of angle JJ in this diagram, we will utilize the following concepts:

  1. Parallel Line Angle Relationships:
    • Alternate interior angles are equal.
    • Linear pairs add up to 180180^\circ.

Step-by-Step Solution:

  1. Expression for Alternate Interior Angles: Since (5x12)(5x - 12)^\circ and (1/2x+38)(1/2x + 38)^\circ are alternate interior angles, they must be equal because lines ff and gg are parallel. Therefore: 5x12=12x+385x - 12 = \frac{1}{2}x + 38

  2. Solve for xx:

    • Subtract 12x\frac{1}{2}x from both sides: 5x12x12=385x - \frac{1}{2}x - 12 = 38 102x12x=50\frac{10}{2}x - \frac{1}{2}x = 50 92x=50\frac{9}{2}x = 50
    • Multiply through by 2 to clear the fraction: 9x=1009x = 100
    • Divide by 9: x=1009x = \frac{100}{9}
  3. Calculate J\angle J: Substitute x=1009x = \frac{100}{9} back into one of the angle expressions, e.g., 5x125x - 12: 5x12=5×100912=5009125x - 12 = 5 \times \frac{100}{9} - 12 = \frac{500}{9} - 12 Convert 1212 to have a denominator of 9: 50091089=3929\frac{500}{9} - \frac{108}{9} = \frac{392}{9} Approximate: 392943.6\frac{392}{9} \approx 43.6^\circ

Thus, the measure of angle JJ is approximately 4343^\circ, aligning with option B.


Would you like further clarification on this solution? Here are 5 related questions to expand your understanding:

  1. Why are alternate interior angles equal when two parallel lines are cut by a transversal?
  2. Can this method be used if the lines were not parallel?
  3. How do we determine when to set angle expressions equal versus summing to 180180^\circ?
  4. How do the properties of linear pairs help solve for angles in similar problems?
  5. What are some other common angle relationships in parallel line diagrams?

Tip: Always double-check which angle relationship applies in the given scenario—mistaking corresponding for alternate interior angles is a common error!

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Math Problem Analysis

Mathematical Concepts

Parallel Lines
Transversals
Angle Relationships
Algebraic Equations

Formulas

5x - 12 = 1/2x + 38
Alternate interior angles are equal
Linear pairs add up to 180°

Theorems

Alternate Interior Angles Theorem
Linear Pair Theorem

Suitable Grade Level

Grade 8