In an isosceles triangle, the measure of one of the three angles is \((80 - 5\mathrm{y})°\). The sum of the...

1. TRANSLATE the problem information

• Given information:
  • We have an isosceles triangle
  • One angle measures \((80 - 5\mathrm{y})°\)
  • \(\mathrm{S}°\) represents the sum of two angles from the triangle
  • Need to find which expression could NOT equal S

2. INFER the key insight about isosceles triangles

• An isosceles triangle has exactly two equal angles (called base angles) and one different angle (the vertex angle)
• All three angles sum to \(180°\)

3. CONSIDER ALL CASES for what the given angle represents

The angle \((80 - 5\mathrm{y})°\) could be either:

• Case 1: One of the base angles (equal angles)
• Case 2: The vertex angle (different angle)

We must analyze both possibilities to find all values S could take.

4. INFER and SIMPLIFY Case 1 calculations

Case 1: \((80 - 5\mathrm{y})°\) is a base angle

• Both base angles = \(80 - 5\mathrm{y}\)
• Vertex angle = \(180 - 2(80 - 5\mathrm{y})\)

\(180 - 2(80 - 5\mathrm{y}) = 20 + 10\mathrm{y}\)

Possible sums of two angles:

• Two base angles: \(2(80 - 5\mathrm{y}) = 160 - 10\mathrm{y}\)
• Base angle + vertex angle: \((80 - 5\mathrm{y}) + (20 + 10\mathrm{y}) = 100 + 5\mathrm{y}\)

5. INFER and SIMPLIFY Case 2 calculations

Case 2: \((80 - 5\mathrm{y})°\) is the vertex angle

• Vertex angle = \(80 - 5\mathrm{y}\)
• Sum of both base angles = \(180 - (80 - 5\mathrm{y}) = 100 + 5\mathrm{y}\)
• Each base angle = \((100 + 5\mathrm{y}) \div 2 = 50 + 2.5\mathrm{y}\)

Possible sums of two angles:

• Two base angles: \(100 + 5\mathrm{y}\)
• Base angle + vertex angle: \((50 + 2.5\mathrm{y}) + (80 - 5\mathrm{y}) = 130 - 2.5\mathrm{y}\)

6. INFER the complete solution set

All possible values for S: \(\{160 - 10\mathrm{y}, 100 + 5\mathrm{y}, 130 - 2.5\mathrm{y}\}\)

Now check each answer choice:

• (A) \(100 + 5\mathrm{y}\) ✓ (appears in our set)
• (B) \(130 - 2.5\mathrm{y}\) ✓ (appears in our set)
• (C) \(130 + 2.5\mathrm{y}\) ✗ (does NOT appear in our set)
• (D) \(160 - 10\mathrm{y}\) ✓ (appears in our set)

Answer: C \((130 + 2.5\mathrm{y})\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often assume the given angle must be a base angle (since isosceles triangles are usually drawn with the vertex angle at the top) and fail to consider it could be the vertex angle.

When they only analyze one case, they miss some possible values of S. This incomplete analysis may lead them to incorrectly conclude that an expression like (C) \(130 + 2.5\mathrm{y}\) could be valid, or they become confused about which expressions are actually possible and end up guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes when calculating the vertex angle from the base angles or when combining expressions to find sums.

For example, they might incorrectly calculate:

\(180 - 2(80 - 5\mathrm{y})\)

\(= 180 - 160 - 10\mathrm{y}\)

\(= 20 - 10\mathrm{y}\)

instead of \(20 + 10\mathrm{y}\). These computational errors lead to wrong possible values for S, causing them to eliminate the wrong answer choice.

The Bottom Line:

This problem requires systematic case analysis - a skill many students struggle with because it demands considering multiple scenarios rather than following a single solution path. The key insight is recognizing that "one of the three angles" doesn't specify which type of angle it is in the isosceles triangle.