Understanding Quadrilaterals - 8th math

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NCERT Exercise 3.3


Understanding Quadrilaterals NCERT Exercise 3.3 Question (1) Given parallelogram ABCD. Complete each statement along with the definition or property used.

understanding quadrilaterals ncert exercise 3.3 question1

(i) AD = …….

(ii) ∠ DCB = …….

(iii) OC = ……..

(iv) m∠DAB + m∠CDA = ……

Solution

(i) AD = BC

Explanation Because opposite sides in a parallelogram are equal.

(ii) ∠ DCB = ∠ DAB

Explanation Because opposite angles in a parallelogram are equal.

(iii) OC = OA

Explanation Because diagonals of a parallelogram bisects one another.

(iv) m∠DAB + m∠CDA = 1800

Explanation Because sum of measure of adjacent angles in a parallelogram is equal to 1800

Understanding Quadrilaterals NCERT Exercise 3.3 Question (2) Consider the following parallelograms. Find the values of the unknowns x, y, z.

(i)

understanding quadrilaterals ncert exercise 3.3 question2

Solution

Given, ABCD is a parallelogram

And, ∠ B = 1000

Thus, x, y and z = ?

In between ∠A and ∠B

Since, adjacent angle of a quadrilateral are supplementary,

Hence ∠A + ∠B = 1800

⇒ z + 1000 = 1800

⇒ z = 1800 – 1000

⇒ z = 800

In between ∠B and ∠D

Both ∠B and ∠D are opposite angles of a parallelogram.

And we know that opposite angles of a parallelogram are equal in measure.

Thus, ∠D = ∠B

⇒ y = 1000

In between ∠A and ∠C

Both ∠A and ∠C are opposite angles of a parallelogram.

And we know that opposite angles of a parallelogram are equal in measure.

Thus, ∠D = ∠B

⇒ z = 800

Thus, x = 800, y = 1000 and z = 800 Answer

(ii)

understanding quadrilaterals ncert exercise 3.3 question2-ii

Solution

understanding quadrilaterals ncert exercise 3.3 question2-ii-answer

Given, ABDE is a parallelogram

And, ∠ E = 500

Thus, x, y and z = ?

Between ∠ E and ∠ABD in the given parallelogram

We know that, opposite angles of a parallelogram are equal.

Thus, ∠ E = ∠ ABD = 500

Between ∠ ABD and z

Since, both the angles ∠ ABD and z form a straight line, and hence is supplementary

Thus, ∠ ABD + z = 1800

⇒ 500 + z = 1800

⇒ z = 1800 – 500

⇒ z = 1300

Now, we know that, sum of measure of internal angles of a quadrilateral = 3600

Thus, in the given quadrilateral

∠E + x + ∠ABD + y = 3600

⇒ 500 + x + 500 + y = 3600

⇒ 1000 + x + x = 3600

[∵ x and y are vertically opposite angles of a parallelogram, and thus are equal. And thus x = y]

⇒ 1000 + 2x = 3600

⇒ 2x = 3600 – 100

⇒ 2x = 2600

`=>x=260^o/2`

⇒ x = 1300

Thus, x = y = 1300

Thus, x = y = z = 1300 Answer

(iii)

understanding quadrilaterals ncert exercise 3.3 question2-iii

Solution

understanding quadrilaterals ncert exercise 3.3 question2-iii-answer

Given, ABCD is a parallelogram.

And, ∠ B = 300

And, ∠ DOA = 900

Thus, x, y and z = ?

Here given, ∠ DOA = 900

Now, since ∠ DOA and ∠ COB are vertically opposite angles, and hence are equal

And thus ∠ DOA = ∠ COB = 900

Now, in triangle COB,

We know that, sum of interior angles of a triangle = 1800

Thus, x + y + 300 = 1800

900 + y + 300 = 1800

1200 + y = 1800

y = 1800 – 1200

y = 600

Now, between ∠DAO and ∠ OCB

DA || CB [∵ opposite sides of the given parallelogram]

And AC a transversal passing through DA and CB

Thus, ∠ DAO = ∠ OCB

[∵ ∠DAO and ∠OCB are pair of corresponding angle]

⇒ z = y = 600

Thus, x = 900, z = y = 600 Answer

(iv)

understanding quadrilaterals ncert exercise 3.3 question2-iv

Solution

understanding quadrilaterals ncert exercise 3.3 question2-iv-answer

Given, ABCD is a parallelogram

And, ∠ B = 800

Thus, x, y and z = ?

Between ∠B and ∠D

Here, since ∠B and ∠D are vertically opposite angles of a parallelogram, and hence are equal

Thus, ∠D = ∠B

⇒ y = 800

Between ∠B and ∠A

Since, ∠B and ∠A are adjacent angles of a parallelogram, and thus are supplementary.

Thus, ∠ A + ∠ B = 1800

⇒ x + 800 = 1800

⇒ x = 1800 – 800

⇒ x = 1000

Between ∠B and ∠DCE

Being the opposite sides of a parallelogram, AB||DC

And, BC is a transversal passing through these parallel lines

Now, ∠B and ∠DCE form a pair of corresponding angle, and thus are equal.

Thus, ∠DCE = ∠B

⇒ z = 800

Thus, x = 1000, and y and z = 800 Answer

(v)

understanding quadrilaterals ncert exercise 3.3 question2-v

Solution

understanding quadrilaterals ncert exercise 3.3 question2-v-answer

Given, ABCD is a parallelogram

And, AC is one of its diagonal

And, ∠ B = 1120

And, ∠DAC = 400

Thus, angles z, y and x = ?

Between ∠B and ∠D

∠B and ∠D are opposite angles of the given parallelogram.

And we know that, opposite angles of a parallelogram are equal in measure.

Thus, ∠ D = ∠ B

⇒ y = 1120

Now, in triangle ADC

We know that, sum of all the three angles of a triangle = 1800

Thus, in triangle ADC

∠DAC + ∠D + ∠ACD = 1800

⇒ 400 + 1120 + x = 1800

⇒ 1520 + x = 1800

⇒ x = 1800 – 1520

⇒ x = 280

Now, In the given parallelogram ABCD

AB||DC and AC is a transversal which is going through these parallel lines

Now, ∠z and ∠x are pair of alternate interior angles, and we know that, pair of alternate interior angles are equal

Thus, ∠z = ∠x = 280

Thus, x = z = 280 and y = 1120 Answer

Understanding Quadrilaterals NCERT Exercise 3.3 Question (3) Can a quadrilateral ABCD be a parallelogram if

(i) ∠D + ∠B = 1800

understanding quadrilaterals ncert exercise 3.3 question3

Solution

Let ABCD is the given quadrilateral

As given, (i) if ∠D + ∠B = 1800

Then ABCD is a parallelogram or not.

If ABCD is a quadrilateral, then ∠ D and ∠ B will the opposite angles

We know that the sum of opposite angles in a quadrilateral = 1800

And opposite angles are equal in a rectangle.

Thus, in the case of sum of opposite angles is equal to 1800

Then, it may be a parallelogram or may not be. Answer

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

In a parallelogram opposite sides are equal.

Here since only one pair of opposite sides are equal, i.e. AB = DC

And other pair of opposite sides AD and BC are not equal

Thus, in this condition, the given quadrilateral is not a parallelogram Answer

(iii) &8736;A = 700 and ∠C = 650

Here, in quadrilateral &8736;A and &8736;C are opposite angles and are not equal in measure.

We know that in a parallelogram, the opposite angles are equal.

Thus, in this condition, the given quadrilateral is not a parallelogram. Answer

Understanding Quadrilaterals NCERT Exercise 3.3 Question (4) Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

Solution

understanding quadrilaterals ncert exercise 3.3 question4

A kite is a quadrilateral in which exactly two opposite angles are of equal measure, but kite is not a parallelogram as opposite sides are not equal. Answer

Understanding Quadrilaterals NCERT Exercise 3.3 Question (5) The measures of two adjacent angles of a parallelogram are in the ratio 3:2. Find the measure of each of the angles of the parallelogram.

Answer

Let the measure of two adjacent angles of the given parallelogram = 3x and 2x

We know that, the sum of measure of two adjacent angles of a parallelogram = 1800

⇒ 3x + 2x = 1800

⇒ 5x = 1800

`=>x=180^o/5`

⇒ x = 360

Thus, 2x = 2 × 360

= 720

And, 3x = 3 × 360

= 1080

Thus, two adjacent angles of the given parallelogram are 1080 and 720 Answer

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