Understanding Quadrilaterals - 8th math

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

(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)

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)

Solution

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 = 130⁰

Thus, x = y = 130⁰

Thus, x = y = z = 130⁰ Answer

(iii)

Solution

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)

Solution

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)

Solution

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

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

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 = 36⁰

Thus, 2x = 2 × 36⁰

= 72⁰

And, 3x = 3 × 36⁰

= 108⁰

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