Explanation
Goal: Start with a right angle (90°). First bisect it to get two equal angles. Then bisect each of those parts again. Find the size of the final smallest parts.
What you need
- Ruler (to draw straight lines)
- Compass (to mark equal arcs)
- Protractor (to measure and check your result)
- Pencil
A. Bisect the right angle (90° → 45° & 45°)
- Draw a right angle ∠AOB. (OA is one arm, OB is the other arm at 90°.)
- Place the compass point at O. Draw a small arc that cuts both arms OA and OB at points C and D.
- Without changing the compass width, place the compass at C and draw an arc inside the angle.
- With the same width, place the compass at D and draw another arc to intersect the previous arc at point E.
- Draw a straight line from O through E. This line OE is the angle bisector of ∠AOB.
- Now the 90° angle is split into two equal angles: ∠AOE and ∠EOB.
Check with a protractor: Each part should read about 45°.
(90^circ div 2 = 45^circ)
B. Bisect each 45° angle again (45° → 22.5° & 22.5°)
We will repeat the same bisecting steps inside one 45° part (say ∠AOE). Do the same for the other part (∠EOB).
- Keep the compass point at O. Draw an arc cutting the two arms of ∠AOE. Mark those cut points as F and G.
- With the same compass width, draw an arc from F inside the angle.
- With the same width, draw another arc from G to intersect the previous arc at H.
- Draw a line from O through H. This line bisects ∠AOE into two equal parts.
- Repeat the same four steps inside the other 45° angle (∠EOB).
Check with a protractor: Each new small angle should read about 22.5°.
(45^circ div 2 = 22.5^circ)
Why this works (in simple words)
- An angle bisector is a line that divides an angle into two equal angles.
- We first bisected 90° into two equal parts (so each is 45°).
- Then we bisected each 45° again, making two equal parts of 22.5° each.
Final calculation (shown in small steps)
(90^circ = ext{right angle})
(90^circ div 2 = 45^circ)
(45^circ div 2 = 22.5^circ)
Answer: After two bisections, the measure of each smallest part is 22.5°.