In an isosceles triangle, the measure of the vertex angle is 48°. What is the measure, in degrees, of one...

1. TRANSLATE the problem information

• Given information:
  • We have an isosceles triangle
  • The vertex angle measures \(48°\)
  • We need to find the measure of one base angle

2. INFER the approach using isosceles triangle properties

• In an isosceles triangle, the two base angles are always equal
• Let \(\mathrm{x}\) = the measure of one base angle
• Since both base angles equal \(\mathrm{x}\), their combined measure is \(2\mathrm{x}\)
• We can use the triangle angle sum property: all angles add to \(180°\)

3. TRANSLATE this into an equation

• vertex angle + base angle + base angle = \(180°\)
• \(48° + \mathrm{x} + \mathrm{x} = 180°\)
• \(48° + 2\mathrm{x} = 180°\)

4. SIMPLIFY by solving for x

• Subtract \(48°\) from both sides: \(2\mathrm{x} = 180° - 48°\)
• \(2\mathrm{x} = 132°\)
• Divide by 2: \(\mathrm{x} = 66°\)

Answer: C (66)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse which angle is the vertex angle versus the base angles in an isosceles triangle. Some students think all three angles are equal (confusing isosceles with equilateral), leading them to divide \(180°\) by 3 to get \(60°\). Since \(60°\) isn't an answer choice, this leads to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the equation correctly (\(2\mathrm{x} + 48 = 180\)) but make arithmetic errors. A common mistake is: \(2\mathrm{x} = 180 + 48 = 228\), then \(\mathrm{x} = 114°\). Since this exceeds \(90°\), some students might select the closest answer choice or realize something's wrong but then guess randomly.

The Bottom Line:

This problem tests whether students truly understand isosceles triangle properties and can distinguish between vertex and base angles. The algebraic setup is straightforward once the geometric relationships are clear.