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) — every question and answer worked out in full, not just the final result. You can also read the Quadrilaterals textbook chapter.

Solved

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…

Sample

Worked examples

3 questions from Quadrilaterals, solved step by step.

Rectangles and Squares

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?

This is a Try This problem. The question asks whether 'opposite sides parallel and equal' is sufficient to define a rectangle.

Answer: NO — this condition alone is not sufficient to define a rectangle.

A parallelogram has opposite sides parallel and equal by definition, but a parallelogram need NOT have right angles.

For example, a rhombus (not a square) has all sides equal (so opposite sides equal) and opposite sides parallel, but its angles are not 90°.

More generally, any non-rectangular parallelogram has opposite sides parallel and equal, yet is not a rectangle.

Therefore, 'opposite sides parallel and equal' describes a PARALLELOGRAM, which is a broader set than rectangles.

A rectangle requires the additional condition that all angles are 90° (or equivalently, that the diagonals are equal).

Answer: No. A quadrilateral with opposite sides parallel and equal is a parallelogram, which need not be a rectangle. An additional condition (all angles = 90°, or diagonals equal) is needed to guarantee a rectangle.

Kite and Trapezium

Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.

Take two equilateral triangles PQR and PQS, each with sides 4 cm, joined along the common side PQ.

The resulting quadrilateral has vertices R, Q, S, P (going around).

Sides:

All four sides of the quadrilateral = the sides of the equilateral triangles = 4 cm each.

So it is a rhombus with all sides = 4 cm.

Angles:

Each interior angle of an equilateral triangle = 60°.

At vertex R (or S): only one triangle contributes ⇒ angle R = angle S = 60°.

At vertex P (or Q): two triangles meet, each contributing 60° on the same side of PQ.

angle at P = angle RPQ + angle SPQ = 60° + 60° = 120°.

angle at Q = angle RQP + angle SQP = 60° + 60° = 120°.

Check: 60° + 120° + 60° + 120° = 360° ✓

The quadrilateral RPSQ (or RQSP) is a rhombus with:

All sides = 4 cm, angles at R and S = 60°, angles at P and Q = 120°.

Answer: The quadrilateral is a rhombus with all sides = 4 cm; angles at R and S = 60°, angles at P and Q = 120°.

Kite and Trapezium

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.)

Rectangle PAIR: P(top-left), A(top-right), I(bottom-right), R(bottom-left).

The vertex O of rectangle RODS sits on the side AI of PAIR, and ∠ORI = 30° is given (angle at R between RO and the base RI).

Step 1: Find ∠ROI in triangle ROI.

Since PAIR is a rectangle, its side AI is perpendicular to its side RI, so ∠OIR = 90° (O lies on AI, directly above I).

In right triangle ROI: ∠ROI = 90° − ∠ORI = 90° − 30° = 60°.

Step 2: Use the right angle of rectangle RODS at O.

In rectangle RODS the vertices go R → O → D → S, so the sides RO and OD meet at O at a right angle: ∠ROD = 90°.

Step 3: Find ∠IOD.

The ray OI lies inside ∠ROD (O–I is between O–R and O–D), so

∠IOD = ∠ROD − ∠ROI = 90° − 60° = 30°.

Therefore ∠IOD = 30°.

Answer: ∠IOD = 30°. (Since AI ⊥ RI in rectangle PAIR, ∠ROI = 90° − 30° = 60°; since ∠ROD = 90° in rectangle RODS, ∠IOD = 90° − 60° = 30°.)

All 19 questions in Quadrilaterals are solved step by step in the Solutions PDF above.

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Chapter 4 Solutions — Quadrilaterals

Overview

What these solutions cover

Rectangles and Squares

More Quadrilaterals with Parallel Opposite Sides (Parallelograms and Rhombuses)

Kite and Trapezium

Worked examples

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?

Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.

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