Class 8 Mathematics

Chapter 4 Solutions — Quadrilaterals

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Solutions

Overview

Step-by-step NCERT solutions for Quadrilaterals (Chapter 4, NCERT Class 8 Mathematics) — the full working for every question, not just the final answer. You can also read the Quadrilaterals textbook chapter.

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What these solutions cover

All 19 questions in Quadrilaterals are solved in the PDF. Here's what's inside, exercise by exercise:

Rectangles and Squares

1.
Find all the other angles inside the following rectangles.
- (i) Rectangle with vertices D(top-left), C(top-right), A(bottom-left), B(bottom-right). The diagonal AC makes an angle of 30° with the base AB at vertex A, i.e. ∠CAB = 30°.
- (ii) Rectangle with vertices Q(top-left), R(top-right), P(bottom-left), S(bottom-right). The diagonals QS and RP meet at O, and the angle at O opening towards the top…
2.
Draw a quadrilateral whose diagonals have equal lengths of 8 cm that bisect each other, and intersect at an angle of
- (i) 30°
- (ii) 40°
- (iii) 90°
- (iv) 140°.
Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.
We have seen how to get 90° using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90° using these?
We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?

More Quadrilaterals with Parallel Opposite Sides (Parallelograms and Rhombuses)

1.
Find the remaining angles in the following quadrilaterals.
- (i) Parallelogram PEAR: ∠P = 40°.
- (ii) Parallelogram SPQR: ∠P = 110°.
- (iii) Rhombus XWVU: the diagonal at V makes 30° with each side (∠XVW = ∠XVU = 30°, so full ∠V = 60°).
- (iv) Rhombus OIAE: the diagonal at E makes 20° with each side (∠OEI = 20°, so full ∠E = 40°).
Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.
Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.

Kite and Trapezium

Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.
Construct a kite whose diagonals are of lengths 6 cm and 8 cm.
3.
Find the remaining angles in the following trapeziums —
- (i) A trapezium with bottom angles 135° and 105°.
- (ii) An isosceles trapezium with one angle of 100°.
4.
Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then answer:
- (i) What is the quadrilateral that is both a kite and a parallelogram?
- (ii) Can there be a quadrilateral that is both a kite and a rectangle?
- (iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?
If PAIR and RODS are two rectangles, find ∠IOD. (In rectangle PAIR: P(top-left), A(top-right), I(bottom-right), R(bottom-left); the left side PR = 5 cm. The vertex O of rectangle RODS lies on the side AI, and RO makes an angle of 30° with the base RI, i.e. ∠ORI = 30°. In rectangle RODS the vertices are R, O, D, S in order, with the side RS = 5 cm.)
Construct a square with diagonal 6 cm without using a protractor.
CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement.
If a quadrilateral has four equal sides and one angle of 90°, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.
What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer. Hint: Draw a diagonal and check for congruent triangles.
Will the sum of the angles in a quadrilateral such as the following one (a non-convex/concave quadrilateral, where one vertex D lies inside triangle ABC) also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring.
11.
State whether the following statements are true or false. Justify your answers.
- (i) A quadrilateral whose diagonals are equal and bisect each other must be a square.
- (ii) A quadrilateral having three right angles must be a rectangle.
- (iii) A quadrilateral whose diagonals bisect each other must be a parallelogram.
- (iv) A quadrilateral whose diagonals are perpendicular to each other must be a…

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