Quadrilaterals l Exercise - 4.3 l Class 8 l NCERT Solutions l Ganita Prakash

Quadrilateral – Complete Solutions

Q1

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

Each triangle has all sides = 4 cm and angles = 60°.

Joining two triangles forms a rhombus.

Sides: All sides = 4 cm

Angles: 60°, 120°, 60°, 120°

Q2

Construct a kite whose diagonals are 6 cm and 8 cm.

Draw AC = 8 cm.
Find midpoint O.
Draw perpendicular at O.
Mark OB = OD = 3 cm.
Join A-B-C-D-A.

Q3

Find the remaining angles in the trapeziums.

Left trapezium:

Angles on same side of parallel lines are supplementary.

Remaining angles = 45° and 75°

Right trapezium:

Using isosceles trapezium property:

Angles = 100°, 80°, 100°, 80°

Q4

Venn diagram relationships between parallelogram, kite, rhombus, rectangle, square.

(i) Both kite and parallelogram → Rhombus

(ii) Yes → Square

(iii) No. Every rhombus is a kite, but not every kite is a rhombus.

Q5

If PAIR and RODS are rectangles, find ∠IOD.

Given angle at R = 30°

Using rectangle properties (right angles):

∠IOD = 60°

Quadrilateral Solutions (Q6–Q11)

Q6: Square Construction

Construct a square with diagonal 6 cm without using a protractor.

Draw AC = 6 cm.
Find midpoint O.
Draw perpendicular at O.
Mark OB = OD = 3 cm.
Join A-B-C-D-A → Required square.

Q7

CASE is a square. U, V, W, X are midpoints. What is UVWX?

Joining midpoints of a square forms another square.

UVWX is a square.

Q8

If a quadrilateral has four equal sides and one right angle, is it a square?

Yes.

Equal sides → Rhombus

One angle = 90° → all angles become 90°

Therefore, it is a square.

Q9

What type of quadrilateral has opposite sides equal?

Parallelogram

Opposite sides equal ⇒ triangles formed by diagonal are congruent.

Q10

Will the sum of angles in this quadrilateral be 360°?

Yes.

Divide into two triangles → each = 180°

Total = 360°

Q11

(i) A quadrilateral whose diagonals are equal and bisect each other must be a square.

False

Example: Rectangle → diagonals equal and bisect each other but not all sides equal.

Quadrilateral Statements – True or False

(ii) A quadrilateral having three right angles must be a rectangle.

If a quadrilateral has three right angles (90° each), then the fourth angle must also be 90° (since sum = 360°). So, all four angles are right angles.

Answer: True

(iii) A quadrilateral whose diagonals bisect each other must be a parallelogram.

This is a defining property of a parallelogram: its diagonals bisect each other.

Answer: True

(iv) A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.

This is not always true. For example, a kite also has perpendicular diagonals but is not a rhombus.

Answer: False

(v) A quadrilateral in which the opposite angles are equal must be a parallelogram.

If both pairs of opposite angles are equal, the quadrilateral is a parallelogram.

Answer: True

(vi) A quadrilateral in which all the angles are equal is a rectangle.

If all angles are equal, each must be 90°, so the quadrilateral is a rectangle.

Answer: True

(vii) Isosceles trapeziums are parallelograms.

An isosceles trapezium has only one pair of parallel sides, while a parallelogram has two pairs.

Answer: False