Chapter 8: Quadrilaterals
A quadrilateral is a four-sided polygon, the next step up from a triangle. With one more side comes greater richness: special families like parallelograms, rectangles, rhombi, squares, trapeziums, and kites, each with its own properties and proofs. This chapter is a guided tour of those families and the theorems that connect them.
The first headline result is the angle sum: the four interior angles of any quadrilateral sum to . The proof takes one auxiliary line , a diagonal , and reduces to two applications of the triangle angle sum.
The bulk of the chapter studies the parallelogram: a quadrilateral with both pairs of opposite sides parallel. Parallelograms have a rich set of properties , opposite sides equal, opposite angles equal, diagonals bisecting each other , and equally rich converses. If you can show any one of these properties, the quadrilateral is a parallelogram. This let us classify and prove things about rectangles, rhombi, and squares as special parallelograms.
The chapter closes with the midpoint theorem: in any triangle, the segment joining the midpoints of two sides is parallel to the third side and half its length. This compact theorem has applications throughout geometry , proving figures are parallelograms, constructing midpoints, dividing segments into equal parts.
By the end you should know each of these named quadrilaterals by sight, know what properties characterise it, and be able to prove that a given quadrilateral is (or isn't) a parallelogram using one of several equivalent conditions.
What's inside
- Angle sum of a quadrilateral , proof and applications.
- The parallelogram , definition and basic properties.
- Special parallelograms , rectangle, rhombus, square.
- Conditions for a parallelogram , five equivalent criteria.
- Midpoint theorem , its statement, proof, and converse.
Key results / Formula card
| Result | Statement |
|---|---|
| Angle sum | The four interior angles of a quadrilateral sum to . |
| Parallelogram , sides | Opposite sides are equal. |
| Parallelogram , angles | Opposite angles are equal. |
| Parallelogram , diagonals | The diagonals bisect each other. |
| Parallelogram , consecutive angles | Consecutive angles sum to . |
| Rectangle | A parallelogram with one right angle (hence all four are right angles). |
| Rhombus | A parallelogram with one pair of adjacent sides equal (hence all four). |
| Square | A rectangle that is also a rhombus. |
| Diagonals of rhombus | Bisect each other at right angles. |
| Diagonals of rectangle | Are equal in length. |
| Diagonals of square | Are equal and bisect each other at right angles. |
| Midpoint theorem | The segment joining midpoints of two sides of a triangle is parallel to the third and half its length. |
| Converse | A line through the midpoint of one side, parallel to a second side, meets the third side at its midpoint. |
Keep this card next to you while working through proofs. Every problem reduces to one or two of these lines.