Special Types of Quadrilaterals
ICSE Class 8 · Mathematics · Chapter 17
What this chapter covers
- The quadrilateral family
- Properties of a parallelogram: sides and angles
- Diagonals of a parallelogram
- The rectangle
- The rhombus
- The square
- Trapezium and kite
- How the properties are linked
Key points
- A quadrilateral has four sides, four angles and two diagonals, and its angles add up to 360°.
- In a parallelogram, opposite sides are parallel and equal, opposite angles are equal, and adjacent angles add up to 180°.
- The diagonals of a parallelogram bisect each other, but they are not equal in general.
- The diagonals of a rectangle are equal and bisect each other.
- The diagonals of a rhombus bisect each other at right angles and bisect the angles at the vertices.
- The diagonals of a square are equal and bisect each other at right angles, making four 45° angles at the corners.
- A trapezium has exactly one pair of parallel sides; angles along each non-parallel side add up to 180°.
- A kite has two pairs of equal adjacent sides; its diagonals meet at right angles and one pair of opposite angles is equal.
- Every square is a rectangle and a rhombus, and every rectangle and rhombus is a parallelogram.
Questions answered
Define a parallelogram and state its three basic properties.
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Its opposite sides are equal, its opposite angles are equal, and its diagonals bisect each other.
One angle of a parallelogram is 48°. Find the other three angles.
The adjacent angle is 180° - 48° = 132°. The four angles are 48°, 132°, 48° and 132°.
The diagonals of a parallelogram are 18 cm and 12 cm. They meet at O. How far is O from each vertex?
O is the midpoint of each diagonal, so it is 9 cm from the ends of the longer diagonal and 6 cm from the ends of the shorter one.
Which quadrilaterals have equal diagonals: parallelogram, rectangle, rhombus, square?
Only the rectangle and the square.
The diagonals of a rhombus are 16 cm and 12 cm. Find its side and perimeter.
Half-diagonals are 8 cm and 6 cm, and they meet at right angles. Side = √(82 + 62) = √(100) = 10 cm. Perimeter = 4 × 10 = 40 cm.
In rectangle ABCD, the diagonals meet at O and OA = 7 cm. Find BD.
AC = 14 cm, and the diagonals of a rectangle are equal, so BD = 14 cm.
Give a reason why the diagonals of a rhombus meet at right angles.
Each end of one diagonal is equally far from the two ends of the other diagonal, because all sides are equal. So that diagonal lies on the line that cuts the other in half at right angles.
In trapezium ABCD with AB ∥ DC, ∠ A = 55°. Find ∠ D.
Angles along a non-parallel side are co-interior, so ∠ D = 180° - 55° = 125°.