Which of the following angles cannot be constructed using a protractor, a compass, and a sharpened pencil?

Analysis of the Question

The question asks us to identify which of the given angles cannot be constructed using only a ruler (straightedge) and a pair of compasses. In geometry, "construction" refers to drawing precise geometric figures using only these two tools.

Concept Explanation

In Euclidean geometry, an angle can be constructed using a compass and straightedge if and only if it can be expressed as a combination of bisections (dividing by 2) and the construction of equilateral triangles (60°).

Specifically:

1. Basic angles: We can easily construct 60° (equilateral triangle), 90° (perpendicular bisector), and 45° (bisecting 90°).
2. Combinations: By adding or bisecting these, we can construct angles like 30°, 15°, 75°, 105°, 120°, 135°, etc.
3. The Rule: Any angle that is a multiple of 3° can be constructed. However, angles that require dividing an angle into parts other than powers of 2 (like 7, 9, or 11 parts) generally cannot be constructed. Specifically, angles that are not multiples of 3° (like 145°) are impossible to construct using only a compass and straightedge.

Step-by-Step Solution

Let's evaluate the options:

• A. 135°: This is $90° + 45°$. Since both 90° and 45° are constructible, 135° is constructible.
• B. 90°: This is the basic perpendicular angle, which is the foundation of most constructions. It is constructible.
• C. 145°: To construct 145°, we would need to be able to construct 5°. Constructing 5° is equivalent to trisecting a 15° angle, which is mathematically proven to be impossible using only a compass and straightedge. Therefore, 145° is not constructible.
• D. 60°: This is the standard angle formed by an equilateral triangle. It is constructible.

Conclusion

The angle 145° cannot be constructed because it is not a multiple of 3° and cannot be derived from the bisection of standard constructible angles.

Exam Tip: Remember that most "constructible" angles in JAMB questions are multiples of 15° (e.g., 15, 30, 45, 60, 75, 90, 105, 120, 135, 150). If you see an angle that isn't a multiple of 15° (or at least 3°), it is highly likely to be the "cannot be constructed" option.