What are the other angles that can be constructed using angle bisection? Can you construct 65.5° angle?

Question

Homi · Accepted Answer

We can create new angles by bisecting existing angles or by adding and subtracting constructible angles.

Step 1 — Constructing basic angles

We can construct a 90-degree angle using a compass and ruler. Bisecting an angle means dividing it into two equal parts. Let us bisect the 90-degree angle. This gives us an angle of 45 degrees.

$90^\circ \div 2 = \mathbf{45^\circ}$

Let us bisect the 45-degree angle. This gives us an angle of 22.5 degrees.

$45^\circ \div 2 = \mathbf{22.5^\circ}$

So, we can construct angles of 45 degrees and 22.5 degrees.

Diagram 1

Step 2 — Constructing other angles by combining

We can add constructible angles together. Let us add 90 degrees and 45 degrees.

$90^\circ + 45^\circ = \mathbf{135^\circ}$

Let us add 90 degrees and 22.5 degrees.

$90^\circ + 22.5^\circ = \mathbf{112.5^\circ}$

Let us add 45 degrees and 22.5 degrees.

$45^\circ + 22.5^\circ = \mathbf{67.5^\circ}$

So, we can also construct angles of 135 degrees, 112.5 degrees, and 67.5 degrees.

Step 3 — Checking for 65.5 degrees

We need to see if 65.5 degrees can be constructed. The angles we found are 45°, 22.5°, 135°, 112.5°, and 67.5°. These angles are all multiples of 22.5 degrees or 11.25 degrees. Let us check if 65.5 degrees is a multiple of 22.5 degrees.

$65.5 \div 22.5 = \frac{131}{45}$

This is not a whole number. So, 65.5 degrees is not a multiple of 22.5 degrees. This means we cannot make 65.5 degrees by combining these angles. Therefore, we cannot construct an angle of 65.5 degrees.

Answer

(i) The other angles that can be constructed using angle bisection are 45°, 22.5°, 135°, 112.5°, and 67.5°. (ii) No, we cannot construct an angle of 65.5°.

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