Four angles, one total

The idea

The sum of all the angles of any quadrilateral is 360°, because a diagonal divides it into two triangles whose angles each add up to 180°.

A fixed total?

A rectangle has four angles of 90°, so they add up to 360°. But most quadrilaterals have four different angles.

If you bend a quadrilateral into another shape, what happens to the total of its four angles?

• It stays the same
• It gets bigger or smaller
• It stays 180°

Drag the corners

Drag the corners to change the shape: squeeze it, stretch it, even push a corner in to make a dent. Add up the four angles. When you are ready, cut the shape along a diagonal.

Two triangles

The sum of all the angles of any quadrilateral is 360°, because a diagonal divides it into two triangles whose angles each add up to 180°.

Cut along AC. Triangle ABC has ∠1 + ∠2 + ∠3 = 180° and triangle ACD has ∠4 + ∠5 + ∠6 = 180°. All six angles together: 180° + 180° = 360°.

And these six angles make up the four angles of the quadrilateral: (∠1 + ∠4) + ∠2 + (∠3 + ∠6) + ∠5.

Notes

The sum of all the angles of any quadrilateral is 360°, because a diagonal divides it into two triangles whose angles each add up to 180°.

Check yourself

Three angles of a quadrilateral are 80°, 95° and 110°. How large is the fourth angle?

Answer: 75 °

The four angles add up to 360°. The three known angles make 80° + 95° + 110° = 285°. The fourth angle is 360° − 285° = 75°.

A quadrilateral with a dent has three angles of 50°, 40° and 60°. How large is the fourth angle, at the dent?

Answer: 210 °

The total is still 360°. The three known angles make 50° + 40° + 60° = 150°, so the dent angle is 360° − 150° = 210°.

Why do the angles of a quadrilateral add up to 360°?

Can a quadrilateral have angles of 90°, 90°, 90° and 100°?

• A diagonal splits it into two triangles, and each has angles adding up to 180° — correct. Yes! 180° + 180° = 360°, and the six angles of the triangles make up the four angles of the quadrilateral.
• Because every corner of a quadrilateral is 90°. Only the corners of a rectangle are all 90°. Other quadrilaterals have other angles, but the total is still 360°.
• Because it has four sides and four corners. Four sides alone do not give the total. It comes from cutting into two triangles of 180° each.
• Yes, any four angles are possible. The four angles of a quadrilateral must add up to 360°.
• No, the total would be 370°, not 360° — correct. Yes! 90° + 90° + 90° + 100° = 370°. With three right angles, the fourth must be 90° too.
• No, a quadrilateral cannot have a 100° angle. A 100° angle is fine. It is this particular set of four angles that is impossible.