A quadrilateral is a closed figure with four sides, four angles and four vertices. The sum of interior angles of any quadrilateral = 360 degrees.
- Types of Quadrilaterals:
- Parallelogram: Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other.
- Rectangle: Parallelogram with all angles = 90 degrees; diagonals are equal and bisect each other.
- Rhombus: Parallelogram with all sides equal; diagonals bisect each other at right angles.
- Square: Rectangle + Rhombus; all sides equal, all angles = 90 degrees; diagonals equal and bisect at right angles.
- Trapezium: Exactly one pair of parallel sides.
- Kite: Two pairs of consecutive sides equal.
Midpoint Theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of it.
Converse of Midpoint Theorem: A line drawn through the midpoint of one side of a triangle parallel to another side bisects the third side.
- 1.Key Properties of a Parallelogram:
- 2.Opposite sides are equal (AB = CD, BC = AD).
- 3.Opposite angles are equal.
- 4.Consecutive angles are supplementary.
- 5.Diagonals bisect each other.
- 6.Each diagonal divides it into two congruent triangles.
ABCD is a parallelogram with angle A = 70 degrees. Find all other angles.
Opposite angle C = 70 degrees. Consecutive angles are supplementary: angle B = angle D = 180 - 70 = 110 degrees.
Show that the diagonals of a rectangle are equal.
In rectangle ABCD: triangles ABC and DCB share BC, AB = DC (opposite sides), angle ABC = angle DCB = 90 degrees. By SAS, triangle ABC is congruent to triangle DCB, so AC = DB.
Diagonals of a rhombus PQRS intersect at O. If PQ = 5 cm and PR = 6 cm, find QS.
Diagonals of a rhombus bisect each other at right angles. PO = 3 cm, QO2 = PQ2 - PO2 = 25 - 9 = 16, QO = 4 cm. QS = 2 · QO = 8 cm.
In triangle ABC, D and E are midpoints of AB and AC. If DE = 4 cm, find BC.
By midpoint theorem, DE is parallel to BC and DE = BC/2. So BC = 8 cm.
The angles of a quadrilateral are in ratio 2:3:5:8. Find each angle.
Sum = 360. Parts = 2+3+5+8 = 18. Angles: 40, 60, 100, 160 degrees.
In parallelogram ABCD, diagonals AC and BD intersect at O. If OA = 3 cm, find AC.
Diagonals of a parallelogram bisect each other, so OC = OA = 3 cm. AC = 6 cm.
ABCD is a rhombus. If angle DAB = 60 degrees, find angle ABC.
Consecutive angles of a parallelogram are supplementary: angle ABC = 180 - 60 = 120 degrees.
Common mistakes
A rectangle is always a parallelogram, but a parallelogram is not always a rectangle. Students often assume all four sides are equal in a rectangle — only a square has both equal sides and right angles. The midpoint theorem applies to triangles, not all polygons.
Summary
Quadrilaterals are classified by their sides, angles and diagonal properties. The midpoint theorem connects triangle midpoints to parallel lines. Parallelogram properties are key: opposite sides equal, opposite angles equal, and diagonals bisect each other.