What are the other angles that can be constructed using angle bisection? Can you construct 65.5° angle?
Angles Constructible by Bisection: Using angle bisection, we can construct angles that are obtained by repeatedly bisecting known angles.
Starting from 60° (constructible via equilateral triangle): We can get 30°, 15°, 7.5°, 3.75°, etc.
Starting from 90°: We can get 45°, 22.5°, 11.25°, etc.
Can 65.5° be constructed? 65.5° = 131/2° 131 is not divisible by 2 ∴ 65.5° cannot be constructed using angle bisection alone.
65.5° cannot be constructed exactly using ruler and compass from the basic constructible angles through bisection alone.
Explanation
The chapter discusses angle bisection as a method to create new angles from existing ones. Angles that can be constructed through repeated bisection are of the form (base angle)/, where the base angle is constructible (like 60°, 90°). Since 65.5° = 131/2°, and 131 is an odd number, it cannot be obtained by repeatedly halving any standard constructible angle. The constructible angles through bisection are limited to specific values.
Solution Steps
Identify angles constructible by bisection from 60°: 30°, 15°, 7.5°, 3.75°, etc.
Identify angles constructible by bisection from 90°: 45°, 22.5°, 11.25°, etc.
Express 65.5° as 131/2°
Check if 131 is divisible by 2
Conclude: 65.5° cannot be constructed using angle bisection alone