Quadrilaterals

36. The Quadrilateral's Secret · Why four corners always add up to 360°

The four interior angles always add up to 360°.

∠A = 90°∠A = 90°∠B = 90°∠B = 90°∠C = 90°∠C = 90°∠D = 90°∠D = 90°∠A + ∠B + ∠C + ∠D = 90° + 90° + 90° + 90° = 360°∠A + ∠B + ∠C + ∠D = 90° + 90° + 90° + 90° = 360°ABCD
The angle sum of a quadrilateral is always 360°. This is because a quadrilateral can be split into two triangles by a diagonal, and each triangle contributes 180°. Drag any vertex — the four interior angles always sum to 360°.

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Selina ICSE: Quadrilaterals

What this lesson covers

Try to break it

Drag any vertex to reshape the quadrilateral. Watch the angle values update — do they always add up to the same total?

How you build it

Make a quadrilateral with a diagonal.

  • Place point A to start the quadrilateral.
  • Place point B.
  • Place point C.
  • Place point D.
  • Draw the side from A to B.
  • Draw the side from B to C.
  • Draw the side from C to D.
  • Draw the side from D to A to close the quadrilateral.
  • Draw diagonal BD to split the quadrilateral into two triangles.

The proof, step by step

Prove that the four angles of a quadrilateral add up to 360°.

  • Draw diagonal BD inside quadrilateral ABCD.
  • Diagonal BD divides ABCD into two triangles: ΔABD and ΔBCD.
  • The sum of angles in ΔABD is 180°.
  • The sum of angles in ΔBCD is also 180°.
  • Adding both: 180° + 180° = 360°. Thus, ∠A + ∠B + ∠C + ∠D = 360°.

Worked example

In a quadrilateral ABCD, ∠A = 80°, ∠B = 100°, and ∠C = 70°. Find the measure of ∠D.

The sum of interior angles of a quadrilateral is always 360°. Subtract the known angles from 360°: 360° - (80° + 100° + 70°) = 360° - 250° = 110°.

  • 80°
  • 90°
  • 100°
  • 110° — correct
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