In triangleXYZ, the measure of angleX is 23° and the measure of angleY is 66°. What is the measure of...

1. TRANSLATE the problem information

• Given information:
  • Triangle XYZ has \(\angle\mathrm{X} = 23°\)
  • Triangle XYZ has \(\angle\mathrm{Y} = 66°\)
  • Need to find: \(\angle\mathrm{Z}\)

2. INFER the approach

• Since we have two angles of a triangle and need the third, we should use the triangle angle sum theorem
• This theorem tells us that all three interior angles must add up to 180°

3. SIMPLIFY using the triangle angle sum theorem

• Set up the equation: \(\angle\mathrm{X} + \angle\mathrm{Y} + \angle\mathrm{Z} = 180°\)
• Substitute known values: \(23° + 66° + \angle\mathrm{Z} = 180°\)
• Add the known angles: \(89° + \angle\mathrm{Z} = 180°\)
• Solve for \(\angle\mathrm{Z}\): \(\angle\mathrm{Z} = 180° - 89° = 91°\)

Answer: C. 91°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize they need to use the given angles but get confused about what operation to perform. Instead of setting up the triangle angle sum equation, they simply add the two given angles together.

This leads them to calculate \(23° + 66° = 89°\) and think this is the measure of \(\angle\mathrm{Z}\), causing them to select Choice B (89°).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the triangle angle sum theorem but make an arithmetic error in the final calculation. They might accidentally add all three numbers \((23° + 66° + 89° = 178°)\) and then think the answer should be close to 180°.

This confusion about the calculation may lead them to select Choice D (179°).

The Bottom Line:

This problem tests whether students truly understand that the triangle angle sum theorem requires adding the three angles to get 180°, not just working with the given angles in isolation. The key insight is recognizing that you're solving for the third angle, not combining the first two.