The measures of the three angles in a triangle are given by x°, \(\mathrm{(x+20)°}\), and \(\mathrm{(x+40)°}\). What is the measure...

1. TRANSLATE the problem information

• Given information:
  • Three angles: \(\mathrm{x°}\), \(\mathrm{(x+20)°}\), and \(\mathrm{(x+40)°}\)
  • Need to find: the measure of the largest angle

2. INFER the approach

• We need to use the angle sum property of triangles
• Set up an equation: the three angles must add to 180°
• Once we find x, we'll need to determine which expression gives the largest value

3. TRANSLATE into mathematical equation

Set up: \(\mathrm{x + (x+20) + (x+40) = 180}\)

4. SIMPLIFY the equation

• Combine like terms: \(\mathrm{x + x + x + 20 + 40 = 180}\)
• This gives us: \(\mathrm{3x + 60 = 180}\)
• Subtract 60 from both sides: \(\mathrm{3x = 120}\)
• Divide by 3: \(\mathrm{x = 40}\)

5. INFER which angle is largest and calculate

• The three angles are: \(\mathrm{x° = 40°}\), \(\mathrm{(x+20)° = 60°}\), \(\mathrm{(x+40)° = 80°}\)
• The largest angle is \(\mathrm{(x+40)° = 40 + 40 = 80°}\)

Answer: C. 80°

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students solve correctly to get \(\mathrm{x = 40}\), but then select Choice A (40°) as their final answer.

They think finding x completes the problem, not realizing the question asks specifically for "the measure of the largest angle." They need to substitute \(\mathrm{x = 40}\) back into the largest expression \(\mathrm{(x+40)°}\) to get 80°.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes when combining like terms or solving the linear equation.

For example, they might incorrectly combine to get \(\mathrm{2x + 60 = 180}\) instead of \(\mathrm{3x + 60 = 180}\), leading to \(\mathrm{x = 60}\). This would make their largest angle \(\mathrm{60 + 40 = 100°}\), but since this isn't an answer choice, it leads to confusion and guessing.

The Bottom Line:

This is a two-step problem where finding x is only the middle step. Students must remember that the question asks for a specific angle measure, not just the value of the variable.