Explore the angles and diagonals of quadrilaterals through geometric reasoning, and discover the angle sum property that governs every four-sided shape.
The sum of the interior angles of any quadrilateral is always 360°. This can be shown by splitting the quadrilateral into two triangles using a diagonal, since each triangle's angles sum to 180°.
A quadrilateral has three angles measuring 80°, 95°, and 110°. Find the fourth angle.
360° − 80° − 95° − 110° = 75°
Why does splitting a quadrilateral with a diagonal prove the 360° angle sum?
The diagonal creates two triangles, each summing to 180°, so together they give 180°+180° = 360°
Different quadrilaterals have different diagonal properties — in a parallelogram, diagonals bisect each other; in a rhombus, they also intersect at right angles.
In a parallelogram, one diagonal is 10 cm long. What can you say about the two parts it's divided into by the other diagonal's intersection?
Each diagonal bisects the other, so each part of the 10 cm diagonal is 5 cm
What additional property do a rhombus's diagonals have beyond bisecting each other?
They intersect at a right angle (90°)
1. A quadrilateral has angles 70°, 100°, and 95°. Find the fourth angle.
2. Three angles of a quadrilateral are equal, each 85°. Find the fourth angle.
3. In a rectangle, are the diagonals equal in length?
4. A quadrilateral's angles are in the ratio 2:3:3:4. Find each angle.
5. Why do a square's diagonals bisect each other at right angles AND are equal in length?