Understanding Quadrilaterals - 8th math
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.
(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=260^o/2`
⇒ 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=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
Reference: