CBSE Class 8 Mathematics Understanding Quadrilaterals Assignments Set 03

Questions carrying 1 Mark each :-

Q.1 What is the sum of the measures of the four angles of a quadrilateral?
Answer: 360°
In simple words: The interior angles of any four-sided polygon always add up to 360 degrees.

Exam Tip: Use the polygon angle sum formula \((n-2) \times 180^\circ\) where \(n = 4\).

Q.2 If the diagonals of a quadrilateral bisect each other at right angle, the quadrilateral is called ..............
Answer: Rhombus (or Kite in special cases, but specifically a rhombus or square)
In simple words: A quadrilateral whose diagonals cross at 90 degrees and cut each other in half is a rhombus.

Exam Tip: Memorize the specific diagonal properties of special parallelograms: rhombus and square diagonals intersect at right angles.

Q.3 The adjacent angles in a parallelogram are.............
Answer: Supplementary (their sum is 180°)
In simple words: Neighboring angles in a parallelogram add up to 180 degrees because consecutive interior angles are supplementary.

Exam Tip: Adjacent angles add to 180°, while opposite angles are equal.

Q.4 State true or false : A pentagon is a concave polygon.
Answer: False
In simple words: A pentagon can be convex or concave; therefore, stating broadly that a pentagon is a concave polygon is false.

Exam Tip: A polygon is concave if at least one interior angle is greater than 180° and some diagonals lie outside the polygon.

Q.5 A regular polygon is both equiangular and..........
Answer: equilateral
In simple words: A regular polygon must have all equal angles (equiangular) and all equal sides (equilateral).

Exam Tip: Regular polygons satisfy both conditions simultaneously, like an equilateral triangle or a square.

Questions carrying 2 Marks each :-

Q.6 The adjacent sides of a parallelogram are in the ratio 3:7 and its perimeter is 100 cm. Find the sides of the parallelogram.
Answer: Let the common ratio be \(x\).
Adjacent sides are \(3x\) and \(7x\).
Perimeter of a parallelogram = \(2 \times (\text{sum of adjacent sides})\)
\(2(3x + 7x) = 100\)
\(2(10x) = 100\)
\(20x = 100\)
\(x = \frac{100}{20} = 5 \text{ cm}\)
Side 1 = \(3 \times 5 = 15 \text{ cm}\)
Side 2 = \(7 \times 5 = 35 \text{ cm}\)
Therefore, the sides of the parallelogram are 15 cm, 35 cm, 15 cm, and 35 cm.
In simple words: Use the perimeter formula with the given ratio variables to find the side lengths.

Exam Tip: Opposite sides of a parallelogram are equal, so the perimeter is twice the sum of any two adjacent sides.

Q.7 The angles of a quadrilateral are in the ratio 2:3:4:6. Find the measure of the angles.
Answer: Let the angles be \(2x\), \(3x\), \(4x\), and \(6x\).
We know the sum of the angles of a quadrilateral is 360°.
\(2x + 3x + 4x + 6x = 360^\circ\)
\(15x = 360^\circ\)
\(x = \frac{360^\circ}{15} = 24^\circ\)
First angle = \(2 \times 24^\circ = 48^\circ\)
Second angle = \(3 \times 24^\circ = 72^\circ\)
Third angle = \(4 \times 24^\circ = 96^\circ\)
Fourth angle = \(6 \times 24^\circ = 144^\circ\)
The angles are 48°, 72°, 96°, and 144°.
In simple words: Divide the total angle sum (360°) according to the given ratio parts.

Q.8 Find the number of sides of a regular polygon whose each exterior angle has a measure of 36°.
Answer: Sum of all exterior angles of any convex polygon = 360°.
Each exterior angle = 36°.
Number of sides \(n = \frac{360^\circ}{\text{Measure of each exterior angle}}\)
\(n = \frac{360^\circ}{36^\circ} = 10\)
Therefore, the polygon has 10 sides.
In simple words: Divide 360 by the size of one exterior angle to find the number of sides.

Exam Tip: This formula works for any regular polygon where exterior angles are equal.

Q.9 Diagonals AC and BD of a rectangle ABCD intersect each other at point O. If OA=3cm, Find AC and BD.
Answer: In a rectangle, diagonals are equal in length and they bisect each other.
Given \(OA = 3 \text{ cm}\).
Since diagonals bisect each other, \(OC = OA = 3 \text{ cm}\).
Length of diagonal \(AC = OA + OC = 3 + 3 = 6 \text{ cm}\).
Since the diagonals of a rectangle are equal, \(BD = AC = 6 \text{ cm}\).
In simple words: Rectangle diagonals are equal and cut each other in half, so doubling OA gives AC, which equals BD.

Exam Tip: Remember two key rectangle properties: diagonals are equal and bisect each other.

Questions carrying 3 marks each :-

Q.10 Find the unknown in the following figures :-
a) Find angles x, y and z if PQRS is a || gm.
b) Find x and y.
Answer: For part (a):
Given PQRS is a parallelogram. In figure (a), angle at R has 50° and Z°. Wait, let us use parallelogram properties:
Opposite angles are equal, and adjacent angles are supplementary.
From figure (a) layout: angle at S is \(z^\circ\), angle at R is 50°, angle at Q is \(x^\circ\), angle at P is \(y^\circ\). Also line QT forms an exterior angle.
Since opposite angles in a parallelogram are equal:
\(z^\circ = x^\circ\)
Adjacent angles are supplementary: \(z^\circ + 50^\circ = 180^\circ \implies z^\circ = 130^\circ\).
Thus, \(x = 130^\circ\).
Also, opposite angle to \(z^\circ\) is \(y^\circ\), so \(y^\circ = 50^\circ\).
Therefore, \(x = 130^\circ\), \(y = 50^\circ\), \(z = 130^\circ\).

For part (b):
Triangle ABC with exterior angle at vertex B equal to \(120^\circ\), and interior angle at A equal to \(50^\circ\).
Exterior angle property of a triangle states that an exterior angle is equal to the sum of its two interior opposite angles.
\(120^\circ = 50^\circ + x^\circ\)
\(x^\circ = 120^\circ - 50^\circ = 70^\circ\).
Also, angles on a straight line or linear pair at vertex B: \(x^\circ + y^\circ = 180^\circ\).
\(70^\circ + y^\circ = 180^\circ \implies y^\circ = 110^\circ\).
Therefore, \(x = 70^\circ\) and \(y = 110^\circ\).
In simple words: Use parallelogram angle rules for part (a) and triangle exterior angle rules for part (b).

Exam Tip: Always identify whether the figure is a parallelogram or a triangle to apply the correct geometric theorem.

Q.11 Lengths of the diagonals AC and BD of a rhombus are 6cm and 8 cm respectively. Find the length of each side of the rhombus.
Answer: Let the diagonals of rhombus ABCD intersect at point O.
Diagonals of a rhombus bisect each other at right angles (90°).
Therefore, \(AO = \frac{AC}{2} = \frac{6}{2} = 3 \text{ cm}\)
and \(BO = \frac{BD}{2} = \frac{8}{2} = 4 \text{ cm}\).
Triangle AOB is a right-angled triangle at O.
Using Pythagoras theorem in triangle AOB:
\(AB^2 = AO^2 + BO^2\)
\(AB^2 = 3^2 + 4^2\)
\(AB^2 = 9 + 16 = 25\)
\(AB = \sqrt{25} = 5 \text{ cm}\)
Since all sides of a rhombus are equal, the length of each side is 5 cm.
In simple words: Half the diagonals form a right triangle with the side of the rhombus; apply Pythagoras theorem to find the side.

Exam Tip: Diagonals of a rhombus divide it into four congruent right-angled triangles.

Multiple Choice Questions carrying 1 mark each:-

Q.12 The sum of the exterior angles of any polygon is:
(a) 180°
(b) 360°
(c) 90°
(d) 60°
Answer: (b) 360°
The sum of exterior angles taken in order for any convex polygon is always equal to 360°.
In simple words: No matter how many sides a polygon has, turning all the way around its exterior totals 360 degrees.

Exam Tip: This rule is universal for all convex polygons regardless of the number of sides.

Q.13 Which of the following is a regular polygon:
(a) Isosceles triangle
(b) Square
(c) Parallelogram
(d) Rhombus
Answer: (b) Square
A square has all four sides equal and all four angles equal (90°), making it a regular quadrilateral.
In simple words: A square is regular because all its sides and angles are equal.

Exam Tip: An equilateral triangle and a square are standard examples of regular polygons.

Q.14 The opposite angles of a................ are equal.
(a) Parallelogram
(b) Rhombus
(c) Trapezium
(d) None of the options
Answer: (a) Parallelogram (Note: also true for rhombus as it is a type of parallelogram, but parallelogram is the defining quadrilateral class)
In a parallelogram, opposite angles are always equal in measure.
In simple words: Opposite angles facing each other in a parallelogram have identical measurements.

Exam Tip: Memorize the core properties of parallelograms: opposite sides are parallel and equal, and opposite angles are equal.