In triangleXYZ, the measure of angleX is 24° and the measure of angleY is 98°. What is the measure of...

1. INFER the mathematical approach

• Given information:
  • \(\angle\mathrm{X} = 24°\)
  • \(\angle\mathrm{Y} = 98°\)
  • Need to find \(\angle\mathrm{Z}\)

• Key insight: Since we have two angles of a triangle and need the third, we can use the triangle angle sum theorem

2. INFER the equation setup

• The triangle angle sum theorem tells us: \(\angle\mathrm{X} + \angle\mathrm{Y} + \angle\mathrm{Z} = 180°\)
• Substituting known values: \(24° + 98° + \angle\mathrm{Z} = 180°\)

3. SIMPLIFY to solve for the unknown angle

• Combine the known angles: \(24° + 98° = 122°\)
• Our equation becomes: \(122° + \angle\mathrm{Z} = 180°\)
• Isolate \(\angle\mathrm{Z}\): \(\angle\mathrm{Z} = 180° - 122° = 58°\)

Answer: A. 58°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students might misunderstand what the triangle angle sum theorem requires them to do. Instead of recognizing that they need to find the missing angle that makes the sum equal 180°, they might simply add the two given angles together.

This leads them to calculate \(24° + 98° = 122°\) and select Choice C (122°), thinking this is somehow the answer rather than recognizing it's just an intermediate step.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equation \(24° + 98° + \angle\mathrm{Z} = 180°\) but make arithmetic errors in the calculation process. They might incorrectly calculate \(24° + 98°\) or make errors when subtracting from 180°.

This can lead them to select Choice B (74°) or cause confusion that leads to guessing.

The Bottom Line:

This problem tests whether students can connect a fundamental geometry theorem to a straightforward algebraic solution. The key challenge is recognizing that finding a missing angle requires using the constraint that all three angles must sum to 180°, not just working with the given information in isolation.