What are the other angles that can be constructed using angle bisection? Can you construct angle?

Question

Homi · Accepted Answer

Understand the Question

Starting from standard constructible angles (such as $90^\circ$ ), bisecting an angle divides it into two equal halves ( $90^\circ \to 45^\circ \to 22.5^\circ \dots$ ).
Additional angles can be constructed by adding or subtracting these bisected angles (e.g., $90^\circ + 45^\circ = 135^\circ$ , $45^\circ + 22.5^\circ = 67.5^\circ$ ).
To determine if an angle like $65.5^\circ$ can be constructed, check if it can be expressed as a valid combination or bisection of known constructible angles.

(i) What are the other angles that can be constructed using angle bisection?

Step 1 · Bisect Basic Angles

Bisecting an angle divides it into two equal halves. Diagram 1

Bisecting $90^\circ$ : $90^\circ \div 2 = 45^\circ$

Bisecting $45^\circ$ : $45^\circ \div 2 = 22.5^\circ$

Thus, angles of $45^\circ$ and $22.5^\circ$ can be constructed.

Step 2 · Combine Constructible Angles

Combining the bisected angles by addition:

$90^\circ + 45^\circ = 135^\circ$

$90^\circ + 22.5^\circ = 112.5^\circ$

$45^\circ + 22.5^\circ = 67.5^\circ$

Thus, angles of $135^\circ$ , $112.5^\circ$ , and $67.5^\circ$ can also be constructed.

Answer

(i) $45^\circ, 22.5^\circ, 67.5^\circ, 112.5^\circ, 135^\circ$

(ii) Can you construct $65.5^\circ$ angle?

Step 1 · Check Feasibility for $65.5^\circ$

The angles constructed by repeatedly bisecting and combining multiples of $90^\circ$ are multiples of $22.5^\circ$ (or $11.25^\circ$ ).

Dividing $65.5^\circ$ by $22.5^\circ$ : $65.5 \div 22.5 = \dfrac{131}{45}$

Since $\dfrac{131}{45}$ is not an integer, $65.5^\circ$ cannot be formed by combining or bisecting these angles.

Answer

(ii) No, an angle of $65.5^\circ$ cannot be constructed.

Common Mistakes

Confusing Similar Angle Values: Confusing $67.5^\circ$ ( $45^\circ + 22.5^\circ$ , which is constructible) with $65.5^\circ$ (which is not constructible).
Assuming Any Decimal Angle is Constructible: Only angles that result from dividing known base angles by powers of $2$ ( $2, 4, 8, \dots$ ) or combining them can be constructed.

Question 10