Fundamental Concepts

21. Complementary Angles · add up to 90

The sum of two complementary angles is always 90 degrees.

OAB∠AOC = 45°∠AOC = 45°∠COB = 45°∠COB = 45°Sum = 90°Sum = 90°C
Two angles are complementary when their sum is exactly 90° (a right angle). Drag C around the upper arc — ∠AOC and ∠COB always add up to 90°.

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Selina ICSE: Fundamental Concepts

What this lesson covers

Try to break it

What happens if we move C really close to A? Does the sum still stay the same?

How you build it

Make two complementary angles.

  • Place vertex O.
  • Place point A to aim the base arm.
  • Draw ray OA (Ray tool: O then A).
  • Use the Perp tool to draw the perpendicular to OA at O — this gives the right angle ∠AOC = 90°.
  • Place point B between ray OA and ray OC.
  • Draw ray OB. ∠AOB and ∠BOC together fill the right angle, so they add up to 90° — they are complementary.

The proof, step by step

Prove that the angles are complementary

  • The large angle AOB is a right angle, measuring exactly 90°.
  • Ray OC divides angle AOB into two smaller parts: angle AOC and angle COB.
  • Therefore, angle AOC + angle COB = angle AOB = 90°.
  • By definition, two angles whose sum is 90° are called complementary angles.

Worked example

If one angle measures 42°, what is the measure of its complementary angle?

Complementary angles add up to 90°. So, 90° - 42° = 48°.

  • 38°
  • 48° — correct
  • 58°
  • 138°
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