Lines m and n are parallel, and a transversal p intersects both lines as shown in the figure.At the intersection...

1. TRANSLATE the problem information

• Given information:
  • Lines m and n are parallel (shown by the tick marks on both lines)
  • Transversal p intersects both lines
  • At line n: the interior angle on the upper side of n and to the left of p is \((2\mathrm{x} + 64)°\)
  • At line m: the interior angle on the lower side of m and to the right of p is \((4\mathrm{x} + 10)°\)
  • We need to find the value of x

2. INFER the geometric relationship

• Look at the positions of the two marked angles:
  • The \((2\mathrm{x} + 64)°\) angle is above line n (interior to the parallel lines) and on the left side of transversal p
  • The \((4\mathrm{x} + 10)°\) angle is below line m (interior to the parallel lines) and on the right side of transversal p
• These angles are alternate interior angles because:
  • Both are between (interior to) the parallel lines
  • They're on opposite (alternate) sides of the transversal
• INFER the key theorem: When parallel lines are cut by a transversal, alternate interior angles are congruent (equal in measure).

3. TRANSLATE the relationship into an equation

• Since the alternate interior angles must be equal:

\((2\mathrm{x} + 64)° = (4\mathrm{x} + 10)°\)

Or simply: \(2\mathrm{x} + 64 = 4\mathrm{x} + 10\)

4. SIMPLIFY to solve for x

• Move all terms with x to one side and constants to the other:
  • Subtract 2x from both sides: \(64 = 2\mathrm{x} + 10\)
  • Subtract 10 from both sides: \(54 = 2\mathrm{x}\)
  • Divide both sides by 2: \(\mathrm{x} = 27\)

5. Verify the solution

• Check by substituting \(\mathrm{x} = 27\) back into both angle expressions:
  • First angle: \(2(27) + 64 = 54 + 64 = 118°\)
  • Second angle: \(4(27) + 10 = 108 + 10 = 118°\) ✓

Both angles equal 118°, confirming our answer is correct.

Answer: 27

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill - Misidentifying the angle relationship: Students may see two angles marked on parallel lines and assume they're corresponding angles, or they might think these are same-side interior angles (which are supplementary, not equal).

If a student thinks these are same-side interior angles, they would set up:

\((2\mathrm{x} + 64) + (4\mathrm{x} + 10) = 180\)
\(6\mathrm{x} + 74 = 180\)
\(6\mathrm{x} = 106\)
\(\mathrm{x} \approx 17.67\)

This doesn't lead to an integer answer, which contradicts the grid-in instruction, causing confusion and potential guessing.

Second Most Common Error:

Poor SIMPLIFY execution - Algebraic manipulation errors: Even after correctly setting up \(2\mathrm{x} + 64 = 4\mathrm{x} + 10\), students might make sign errors or incorrectly combine terms:

Common mistakes include:

• Subtracting incorrectly: Getting \(2\mathrm{x} + 54 = 4\mathrm{x}\) instead of \(54 = 2\mathrm{x}\)
• Sign errors when moving terms: Getting \(-2\mathrm{x} = 64 - 10 = 54\), then \(\mathrm{x} = -27\)
• Arithmetic errors: Getting \(2\mathrm{x} = 64\) instead of \(2\mathrm{x} = 54\)

These algebraic errors lead to wrong values of x, which wouldn't verify when substituted back into the original expressions.

The Bottom Line:

This problem requires both geometric reasoning (recognizing alternate interior angles) and algebraic skill (solving linear equations accurately). The geometric identification is the critical first step - without correctly identifying the angle relationship, students cannot set up the right equation. Once the equation is established, careful algebraic manipulation is essential to avoid calculation errors.