In triangle JKL, the measure of angle J is 65° and the measure of angle K is 45°. Point M...

1. TRANSLATE the problem information

• Given information:
  • Triangle JKL with \(\mathrm{angle\ J = 65°}\) and \(\mathrm{angle\ K = 45°}\)
  • Point M lies on extension of JL beyond L
  • Exterior angle \(\mathrm{KLM = (5x)°}\)
  • Need to find: value of x

2. INFER which theorem applies

• This is an exterior angle problem - we have an exterior angle and need to relate it to interior angles
• The Exterior Angle Theorem is the key: exterior angle = sum of remote interior angles
• Remote interior angles are J and K (the two angles that don't touch point L)

3. TRANSLATE the theorem into an equation

• Exterior Angle Theorem gives us:
  Measure of angle KLM = Measure of angle J + Measure of angle K
• Substituting known values:
  \(\mathrm{(5x)° = 65° + 45°}\)

4. SIMPLIFY to solve for x

• First, add the right side: \(\mathrm{(5x)° = 110°}\)
• Divide both sides by 5: \(\mathrm{x = 22}\)

Answer: B. 22

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing which theorem applies or confusing interior vs exterior angles

Students might think they need to find the third angle of the triangle first (using the fact that angles sum to 180°), then try to work with that. This leads to unnecessary work and potential confusion about which angles to use.

This may lead them to select Choice D (110) by mistakenly thinking the exterior angle equals 110° directly.

Second Most Common Error:

Poor TRANSLATE execution: Setting up the wrong equation due to misidentifying the remote interior angles

Some students might incorrectly think the exterior angle equals just one of the interior angles, or they might confuse which angles are "remote" to the exterior angle.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

The key insight is recognizing that this is fundamentally an exterior angle theorem problem. Once you identify that angle KLM is exterior and angles J and K are the remote interior angles, the rest becomes straightforward algebra.