A mountain peak forms an isosceles triangle when viewed in cross-section, with the horizontal ground forming the base of the...

1. TRANSLATE the problem information

• Given information:
  • Mountain cross-section forms a triangle
  • Two slopes of equal length meet at summit
  • Angle between slopes at summit = \(74°\)
  • Horizontal ground forms the base
  • Need: angle each slope makes with ground

2. INFER the triangle type and relationships

• Since two slopes have equal length, this is an isosceles triangle
• In isosceles triangles, the base angles (where equal sides meet the base) are equal
• The \(74°\) angle is at the vertex (summit), not at the base

3. INFER the solution strategy

• Let \(\mathrm{x}\) = angle each slope makes with horizontal ground
• Use triangle angle sum: vertex angle + base angle + base angle = \(180°\)

4. SIMPLIFY to solve the equation

• Set up equation: \(74° + \mathrm{x} + \mathrm{x} = 180°\)

• Combine like terms: \(74° + 2\mathrm{x} = 180°\)

• Subtract \(74°\): \(2\mathrm{x} = 106°\)

• Divide by 2: \(\mathrm{x} = 53°\)

Answer: 53

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about triangle types: Students may not recognize this as an isosceles triangle, missing that "two slopes of equal length" creates equal base angles.

Without this recognition, they might try to split the \(74°\) angle in half (thinking each base angle is \(37°\)) or become confused about which angles are equal. This leads to confusion and guessing.

Second Most Common Error:

Weak TRANSLATE skill: Students might misinterpret which angle is \(74°\) - thinking it's the angle between the slope and ground rather than the angle between the two slopes at the summit.

This incorrect setup leads them to solve: \(74° + 74° + \mathrm{x} = 180°\), giving \(\mathrm{x} = 32°\). Since 32 isn't among typical answer choices, this causes them to get stuck and guess.

The Bottom Line:

This problem requires recognizing the geometric setup (isosceles triangle) and correctly identifying which angle measurement goes where. The arithmetic is straightforward once the relationships are clear.