A ladder leans against a vertical wall, forming a right triangle with the ground. The ladder makes a 51° angle...

1. TRANSLATE the problem information

• Given information:
  • Ladder leans against vertical wall
  • Forms right triangle with ground
  • Ladder makes \(51°\) angle with ground
  • Need to find angle ladder makes with wall

2. INFER the geometric setup

• The wall (vertical) and ground (horizontal) meet at \(90°\)
• This creates a right triangle with the ladder as the hypotenuse
• In this right triangle, we have:
  • One right angle (\(90°\)) where wall meets ground
  • Two acute angles: ladder-to-ground (\(51°\)) and ladder-to-wall (unknown)

3. INFER the angle relationship

• In any right triangle, the two acute angles are complementary
• This means: \(\mathrm{acute\ angle\ 1} + \mathrm{acute\ angle\ 2} = 90°\)
• Therefore: \(51° + \mathrm{(ladder\text{-}to\text{-}wall\ angle)} = 90°\)

4. SIMPLIFY to find the answer

• Ladder-to-wall angle = \(90° - 51°\)
  = \(39°\)

Answer: A (\(39°\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize this as a right triangle situation or don't understand that the two acute angles must be complementary.

Instead, they might try to use complex trigonometry or think the answer should be \(51°\) (same as the given angle). This may lead them to select Choice C (\(51°\)) by incorrectly assuming both angles are equal.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand which angles are being described and might confuse the angle with the wall versus the angle with the ground.

This confusion about the geometric setup leads to uncertainty about which calculation to perform, causing them to get stuck and guess.

The Bottom Line:

This problem tests whether students can visualize the geometric relationship and recognize that right triangles have complementary acute angles. The key insight is seeing this as a simple complementary angle problem, not a complex trigonometry situation.