The quadrilateral shown has interior angles measuring m°, 110°, 85°, and n°. If m = 73°, what is the value...

1. TRANSLATE the problem information

• Given information:
  • The quadrilateral has four interior angles: \(\mathrm{m°}\), \(\mathrm{110°}\), \(\mathrm{85°}\), and \(\mathrm{n°}\)
  • We know that \(\mathrm{m = 73°}\)
  • We need to find n
• What we're looking for: The value of the fourth angle, n

2. TRANSLATE using the key geometric property

• For any quadrilateral (four-sided polygon), the sum of all interior angles equals \(\mathrm{360°}\)
• This gives us the equation:
  \(\mathrm{m + 110 + 85 + n = 360}\)

3. SIMPLIFY by substituting the known value

• We know \(\mathrm{m = 73°}\), so substitute this into our equation:
  \(\mathrm{73 + 110 + 85 + n = 360}\)

4. SIMPLIFY by combining the known angles

• Add the three known angle measures:
  \(\mathrm{73 + 110 = 183}\)
  \(\mathrm{183 + 85 = 268}\)
• Our equation becomes:
  \(\mathrm{268 + n = 360}\)

5. SIMPLIFY to isolate and solve for n

• Subtract 268 from both sides:
  \(\mathrm{n = 360 - 268}\)
  \(\mathrm{n = 92}\)

Answer: B (92)

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion: Mixing up polygon angle sum formulas

Some students confuse the quadrilateral angle sum \(\mathrm{(360°)}\) with the triangle angle sum \(\mathrm{(180°)}\). They might think that two angles should be supplementary or use the wrong total:

• If they calculate: \(\mathrm{180 - 73 = 107}\)

This may lead them to select Choice C (107)

Second Most Common Error:

Weak SIMPLIFY execution: Arithmetic error when adding the known angles

The problem requires adding three numbers: \(\mathrm{73 + 110 + 85}\). If a student:

• Misreads \(\mathrm{110°}\) as \(\mathrm{90°}\) (or makes another reading error)
• Calculates incorrectly: \(\mathrm{73 + 90 + 85 = 248}\)
• Then computes: \(\mathrm{360 - 248 = 112}\)

This may lead them to select Choice D (112)

The Bottom Line:

This problem tests whether students remember the correct angle sum formula for quadrilaterals (not triangles) and can accurately perform multi-step arithmetic. The key is knowing that four-sided figures have interior angles totaling \(\mathrm{360°}\), then carefully executing the algebra and arithmetic to isolate the unknown angle.