Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]
239. The Four-Sided Promise · Angles always add up to 360°
The sum of the internal angles of any simple quadrilateral is 360°.
The interior angles of any quadrilateral add up to 360°. A diagonal divides the quadrilateral into two triangles, and since the angles of each triangle total 180°, together they give 2 × 180° = 360°.
What this lesson covers
Try to break it
Drag A, B, C, or D to reshape the quadrilateral. The four interior angles always sum to exactly 360°. Stretch it tall, flat, or even concave — the total snaps back to 360° every time.
How you build it
Make a quadrilateral.
- Draw a line segment AB.
- Draw a line segment BC.
- Draw a line segment CD.
- Draw a line segment DA to close the quadrilateral.
The proof, step by step
Prove that the interior angles of a quadrilateral add up to 360°.
- Draw diagonal AC, dividing the quadrilateral into two triangles: ABC and ADC.
- The sum of angles in triangle ABC is 180°.
- The sum of angles in triangle ADC is 180°.
- Adding them together: 180° + 180° = 360°.
Worked example
In a quadrilateral ABCD, angle A = 80°, angle B = 100°, and angle C = 120°. What is the measure of angle D?
The sum of angles in a quadrilateral is 360°. So, angle D = 360° - (80° + 100° + 120°) = 360° - 300° = 60°.
- 60° — correct
- 80°
- 100°
- 120°