Ex.3.3 Q3 Understanding Quadrilaterals Solution-Ncert Maths Class 8

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Question

Can a quadrilateral \(ABCD\) be a parallelogram if

(i) \(∠D +∠B = 180^\circ?\)

(ii) \(AB = DC = 8 \rm \,cm, AD = 4 \rm \,cm \)and \(BC = 4.4 \rm \,cm?\)

 (iii) \(∠A = 70^\circ\) and \(∠C = 65^\circ?\)

Text Solution

i) What is Known?

Given figure is a quadrilateral

What is Unknown?

If \(ABCD\) is a parallelogram when \(\angle {\rm{D }} + \angle {\rm{B }} = {\rm{18}}0^\circ ?\)

Reasoning:

A parallelogram is a quadrilateral whose opposite sides are parallel.

In parallelogram opposite angles are equal and adjacent angles are supplementary.

Using this property, we can calculate the unknown angles.

Steps:

Using the angle sum property of a quadrilateral,

\[\begin{align}\angle A+\angle B+\angle D+\angle C&={{360}^{\circ}}  \\\angle A+\angle C+{{180}^{\circ}}&={{360}^{\circ}}  \\\angle A+\angle C&={{360}^{\circ}}-{{180}^{\circ}}  \\\
 \angle A+\angle C&={{180}^{\circ}}\text{ (Opposite angles should also be of same measures}\text{.) }\\\end{align}\]

For \(\angle {\rm{D}} + \angle {\rm{B}}\) \(= {\rm{18}}0^\circ \), is a parallelogram.

If the following conditions is fulfilled, then \(ABCD\) is a parallelogram.

The sum of the measures of the adjacent angles should be \(180^\circ\).

Opposite angles should also be of same measure.

ii) What is Known?

Given figure is a quadrilateral.

What is Unknown?

\(ABCD\) be a parallelogram if \(AB = DC = 8 \rm \,cm, AD = 4 \rm \,cm\) and \(BC = 4.4 \rm \,cm\)

Reasoning:

A parallelogram is a quadrilateral whose opposite sides are parallel.

Steps:

Property of parallelogram: The opposite sides of a parallelogram are of equal length. Opposite sides \(AD\) and \(BC \) are of different lengths. So, it’s not parallelogram.

iii) What is Known?

Given figure is a quadrilateral.

What is Unknown?

\(ABCD\) be a parallelogram if \(\angle {\rm{A }} = {\rm{ 7}}0^\circ {\rm{ }}\) and \(\angle {\rm{C }} = {\rm{ 65}}^\circ ?\)

Reasoning:

A parallelogram is a quadrilateral whose opposite sides and angles are equal.

Steps:

Property: In a parallelogram opposite angles are equal.

So, \(\angle {\rm{A }} = {\rm{ 7}}0^\circ {\rm{ }}\)and \( \angle {\rm{C }} = {\rm{ 65}}^\circ {\rm{ }}\)are not equal.

So \(ABCD\) is not parallelogram.