In the figure above, line l is parallel to line m.What is the measure, in degrees, of the angle marked...

1. TRANSLATE the problem information

Looking at the diagram:

• Line l and line m are parallel (\(\mathrm{l \parallel m}\))
• A transversal (the diagonal line) crosses both parallel lines
• One angle is labeled \(\mathrm{(5x - 5)°}\) at the intersection with line l
• Another angle is labeled \(\mathrm{(3x + 15)°}\) at the intersection with line m
• We need to find the measure of the angle \(\mathrm{(3x + 15)°}\)

2. INFER the angle relationship

The key insight: Look at where these angles are positioned relative to the transversal and the parallel lines.

• The angle \(\mathrm{(5x - 5)°}\) is between the parallel lines, on one side of the transversal
• The angle \(\mathrm{(3x + 15)°}\) is between the parallel lines, on the opposite side of the transversal

These are alternate interior angles.

When two parallel lines are cut by a transversal, the Alternate Interior Angles Theorem tells us these angles must be equal in measure.

Therefore: \(\mathrm{5x - 5 = 3x + 15}\)

3. SIMPLIFY to solve for x

Now we solve this linear equation:

\(\mathrm{5x - 5 = 3x + 15}\)

Subtract 3x from both sides:
\(\mathrm{2x - 5 = 15}\)

Add 5 to both sides:
\(\mathrm{2x = 20}\)

Divide by 2:
\(\mathrm{x = 10}\)

4. TRANSLATE back to find the requested angle

The question asks for the measure of the angle marked \(\mathrm{(3x + 15)°}\).

Substitute \(\mathrm{x = 10}\):
\(\mathrm{3(10) + 15 = 30 + 15 = 45}\)

Verification: Let's check the other angle to confirm our work:
\(\mathrm{5(10) - 5 = 50 - 5 = 45}\) ✓

Both angles equal 45°, which confirms they are congruent alternate interior angles.

Answer: 45

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing the angle relationship from the diagram.

Students may look at the diagram and see two angles with different algebraic expressions, but fail to identify that they are alternate interior angles. Without this crucial insight, they don't know that these angles must be equal, so they can't set up the equation \(\mathrm{5x - 5 = 3x + 15}\).

Some students might think the angles are supplementary (adding to 180°) instead of congruent, leading to the equation:
\(\mathrm{(5x - 5) + (3x + 15) = 180}\)
\(\mathrm{8x + 10 = 180}\)
\(\mathrm{8x = 170}\)
\(\mathrm{x = 21.25}\)

Then substituting: \(\mathrm{3(21.25) + 15 = 78.75}\)

This leads to confusion since this doesn't match common angle measures, and the student may abandon the systematic solution and guess.

Second Most Common Error:

Poor TRANSLATE reasoning: Solving for x correctly but substituting into the wrong expression.

A student might correctly identify the alternate interior angles relationship, set up and solve the equation to get \(\mathrm{x = 10}\), but then get confused about which angle the question asks for. They might calculate:
\(\mathrm{5(10) - 5 = 45}\)

While this happens to give the correct answer (since both angles equal 45°), in other problems this type of error would lead to selecting a wrong answer. The question specifically asks for "the angle marked with the expression \(\mathrm{(3x + 15)°}\)."

The Bottom Line:

This problem tests whether students can recognize geometric relationships from a diagram and apply the appropriate theorem. The visual reasoning step—identifying alternate interior angles—is what allows the rest of the solution to proceed. Without strong inference skills connecting the diagram to the relevant theorem, students cannot transform the geometric setup into a solvable algebraic equation.