An isosceles triangle has one unique angle with measure a and two congruent angles each with measure b. The measure...

1. TRANSLATE the problem information

• Given information:
  • Isosceles triangle has one unique angle 'a' and two congruent angles 'b'
  • \(\mathrm{a = 8x + 20}\) degrees
  • \(\mathrm{3a + 2b = 16x + k}\) degrees
  • Need to find the value of k

2. INFER the strategic approach

• Since we have an isosceles triangle, we can use the triangle angle sum property
• The key insight: We need to eliminate one of the variables (either 'a' or 'b') to solve for k
• Strategy: Use \(\mathrm{a + 2b = 180°}\) to express 2b in terms of a, then substitute

3. Apply triangle angle sum property

• For any triangle: \(\mathrm{a + 2b = 180°}\)
• Solving for 2b: \(\mathrm{2b = 180 - a}\)

4. SIMPLIFY by substituting into the given expression

• We know: \(\mathrm{3a + 2b = 16x + k}\)
• Substitute \(\mathrm{2b = 180 - a}\):
  \(\mathrm{3a + (180 - a) = 16x + k}\)
• Combine like terms: \(\mathrm{2a + 180 = 16x + k}\)

5. SIMPLIFY further by substituting the expression for angle 'a'

• We know \(\mathrm{a = 8x + 20}\), so:
  \(\mathrm{2(8x + 20) + 180 = 16x + k}\)
• Distribute: \(\mathrm{16x + 40 + 180 = 16x + k}\)
• Combine constants: \(\mathrm{16x + 220 = 16x + k}\)

6. INFER the final answer

• Since \(\mathrm{16x + 220 = 16x + k}\), we can see that \(\mathrm{k = 220}\)

Answer: D) 220

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may not recognize that they need to use the triangle angle sum property to eliminate variables. Instead, they might try to work directly with the given expressions without establishing the fundamental relationship \(\mathrm{a + 2b = 180°}\).

Without this key insight, they get stuck trying to solve for k with too many unknowns, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the strategy but make algebraic errors during substitution. Common mistakes include:

• Forgetting to distribute the 2 when calculating \(\mathrm{2(8x + 20)}\)
• Arithmetic errors when combining 40 + 180
• Not properly canceling the 16x terms on both sides

This may lead them to select Choice A (140) or Choice C (200) due to calculation errors.

The Bottom Line:

This problem tests whether students can recognize that seemingly complex algebraic expressions in geometry problems often simplify dramatically when you apply fundamental geometric relationships first. The triangle angle sum property is the key that unlocks the entire solution.