Rectilinear Figures
ICSE Class 9 · Mathematics · Chapter 13
What this chapter covers
- Rectilinear figures and quadrilaterals
- The family of special quadrilaterals
- Opposite sides of a parallelogram are equal
- Opposite angles of a parallelogram are equal
- One pair of opposite sides equal and parallel
- Diagonals of a parallelogram
- How to show that a quadrilateral is a parallelogram
- The rhombus: diagonals at right angles
- Rectangles and squares: equal diagonals
- Common mistakes to avoid
Key points
- A rectilinear figure is bounded only by straight line segments; a quadrilateral has four sides, four angles and two diagonals, and its angles add up to 360°.
- Parallelogram: both pairs of opposite sides parallel. Rectangle: a parallelogram with a right angle. Rhombus: a parallelogram with two equal adjacent sides. Square: a rectangle with two equal adjacent sides. Trapezium: exactly one pair of parallel sides.
- In a parallelogram, both pairs of opposite sides are equal (quoted without proof) and both pairs of opposite angles are equal (proved using co-interior angles); adjacent angles add up to 180°.
- A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram (quoted without proof).
- The diagonals of a parallelogram bisect each other, and each diagonal divides it into two congruent triangles of equal area.
- The diagonals of a rhombus bisect each other at right angles and bisect its vertex angles.
- The diagonals of a rectangle are equal; those of a square are equal and bisect each other at right angles.
- To prove a quadrilateral is a parallelogram, show any one of: both pairs of opposite sides parallel, both pairs equal, both pairs of opposite angles equal, one pair equal and parallel, or diagonals bisecting each other.
Questions answered
Define a parallelogram and a rhombus.
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. A rhombus is a parallelogram in which two adjacent sides are equal, so that all four sides are equal.
Three angles of a quadrilateral are 75°, 105° and 120°. Find the fourth angle.
360° - (75° + 105° + 120°) = 360° - 300° = 60°.
One angle of a parallelogram is 48°. Find the other three angles, giving a reason for each.
The opposite angle is 48° (opposite angles of a parallelogram are equal). Each adjacent angle is 180° - 48° = 132° (adjacent angles are supplementary). The angles are 48°, 132°, 48° and 132°.
Two adjacent angles of a parallelogram are in the ratio 2 : 3. Find all its angles.
The adjacent angles add up to 180°, so they are (2)/(5) × 180° = 72° and (3)/(5) × 180° = 108°. The angles are 72°, 108°, 72° and 108°.
In trapezium ABCD, AB ∥ DC, ∠ A = 65° and ∠ B = 50°. Find ∠ D and ∠ C.
AD is a transversal of the parallel sides, so ∠ D = 180° - 65° = 115° (co-interior angles). Similarly ∠ C = 180° - 50° = 130°.
The diagonals of parallelogram PQRS meet at O. If PO = 6.5 cm and QS = 9 cm, find PR and OQ.
The diagonals bisect each other, so PR = 2 × 6.5 = 13 cm and OQ = (1)/(2) × 9 = 4.5 cm.
The side of a rhombus is 13 cm and one diagonal is 24 cm. Find the other diagonal.
Half of the given diagonal is 12 cm. The half-diagonals and a side form a right-angled triangle, so the other half-diagonal is √(132 - 122) = √(25) = 5 cm. The other diagonal is 10 cm.
A square has diagonal 10√(2) cm. Find its side and the angle each diagonal makes with a side.
If the side is a, Pythagoras theorem gives 2a2 = (10√(2))2 = 200, so a = 10 cm. A diagonal bisects a right angle, so it makes 45° with each side it meets.
Past board exam questions
ICSE 2019 sample paper · 3 marks
Each interior angle of a regular polygon is 135°. Find: (i) the measure of each exterior angle. (ii) number of sides of the polygon. (iii) name the polygon.
Show answer
(i) Each exterior angle = 180° − 135° = 45°.
(ii) Number of sides = (360°)/(45°) = 8.
(iii) The polygon is a regular octagon.
ICSE 2019 sample paper · 3 marks
Prove that opposite angles of a parallelogram are equal.
Show answer
Given: ABCD is a parallelogram. To prove: ∠A = ∠C and ∠B = ∠D.
Construction: join AC.
In △ABC and △CDA:
∠BAC = ∠DCA (alternate angles, AB ∥ DC)
∠BCA = ∠DAC (alternate angles, BC ∥ AD)
AC = CA (common)
∴ △ABC ≅ △CDA (ASA), so ∠B = ∠D (c.p.c.t.).
Also ∠A = ∠BAC + ∠DAC = ∠DCA + ∠BCA = ∠C.
Hence opposite angles of a parallelogram are equal.