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# NCERT Solutions(Part- 3)- Understanding Quadrilaterals Class 8 Notes | EduRev

## Mathematics (Maths) Class 8

Created by: Full Circle

## Class 8 : NCERT Solutions(Part- 3)- Understanding Quadrilaterals Class 8 Notes | EduRev

The document NCERT Solutions(Part- 3)- Understanding Quadrilaterals Class 8 Notes | EduRev is a part of the Class 8 Course Mathematics (Maths) Class 8.
All you need of Class 8 at this link: Class 8

EXERCISE 3.3

Question 1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.
(ii) âˆ DCB = _____
(iii) OC = _____
(iv) mâˆ DAB + mâˆ CDA = _____

Solution:
(i) AD = BC               [âˆµ Opposite sides are equal]
(ii) âˆ DCB = âˆ DAB   [âˆµ Opposite angles are equal]
(iii) OC = OA            [âˆµ Diagonals bisect each other]
(iv)mâˆ DAB + mâˆ CDA = 180Â°  [âˆµ Adjacent angles are supplementary]

Question 2. Consider the following parallelograms. Find the values of the unknowns x, y, z.

Solution: (i) âˆ y = 100Â°  [âˆµ Opposite angles of a parallelogram are equal.]
âˆµ Sum of interior angles of a parallelogram = 360Â°
âˆ´  x + y + z + âˆ B = 360Â°
or  x + 100Â° + z + 100Â° = 360Â°
or  x + z = 360Â° â€“ 100Â° â€“ 100Â° = 160Â°

But   x = z
âˆ´ x = z = 160Â°/2 = 80Â°
Thus,

(ii) âˆµ Opposite angles are equal.

âˆ´ mâˆ 1 = 50
Now, âˆ 1 + z = 180Â°  [Linear pair]
or  z = 180Â° â€“ âˆ 1 = 180Â° â€“ 50Â° = 130Â°
x + y + 50Â° + 50Â° = 360Â°

or  x + y = 360Â° â€“ 50Â° â€“ 50Â° = 260Â°
But  x = y
âˆ´  x = y = 260Â°/2Â° = 130Â°

(iii) âˆµ Vertically opposite angles are equal,

âˆ´ x = 90Â°

âˆµ Sum of the angles of a triangle = 180Â° âˆ´90Â° + 30Â° + y = 180Â°
or y = 180Â° â€“ 30Â° â€“ 90Â° = 60Â°

In the figure, ABCD is a parallelogram.
âˆ´ AD || BC and BD is a transversal.
âˆ´ y = z   [Alternate angles]
But y = 60Â°
âˆ´ z = 60Â°
Thus, x = 90Â°, y = 60Â° and z = 60Â°.
(iv) ABCD is a parallelogram.

âˆ´ Opposite angles are equal.
âˆ´ y = 80Â°

AB || CD and BC is a transversal.
âˆ´ x + 80Â° = 180Â° [Interior opposite angles]
or   x = 180Â° â€“ 80Â° = 100Â°

Again BC || AD and CD is a transversal,
âˆ´ z = 80Â°   [Corresponding angles]

Thus, x = 100Â°, y = 80Â° and z = 80Â°
(v) âˆµ In a parallelogram, opposite angles are equal.

âˆ´  y = 112Â°
In Î” ACD,
x + y + 40Â° = 180Â°
x + 112Â° + 40Â° = 180Â°
âˆ´  x = 180Â° â€“ 112Â° â€“ 40Â° = 28Â°
âˆµ AD || BC and AC is a transversal.
âˆ´  x = z    [âˆµ Alternate angles are equal]

and z = 28Â°
Thus, x = 28Â°, y = 112Â° and z = 28Â°

Question 3. Can a quadrilateral ABCD be a parallelogram if
(i) âˆ D + âˆ B = 180Â°?
(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
(iii) âˆ A = 70Â° and âˆ C = 65Â°?
Solution: (i) In a quadrilateral ABCD,

âˆ A + âˆ B = Sum of adjacent angles = 180Â°
âˆ´  The quadrilateral may be a parallelogram but not always.

AB = DC = 8 cm
BC = 4.4 cm
âˆµ Opposite sides AB and BC are not equal.
âˆ´  It cannot be a parallelogram.

âˆ A = 70Â° and âˆ C = 65Â°
âˆµ Opposite angles âˆ A â‰  âˆ C
âˆ´ It cannot be a parallelogram.

Question 4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.
Solution:

In the adjoining figure, ABCD is not a parallelogram such that opposite angles â€“B and â€“D are equal. It is a kite.

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.
Solution: Let ABCD be a parallelogram in which adjacent angles âˆ A and âˆ B are 3x and 2x respectively, since adjacent angles are supplementary.
âˆ´  âˆ A + âˆ B = 180Â°
âˆ´  3x + 2x = 180Â°

or  5x = 180Â°
or   x = 180Â°/5Â° = 36Â°
âˆ´  âˆ A = 3 x 36Â° = 108Â°
and âˆ B = 2 x 36Â° = 72Â°
âˆµ Opposite angles are equal.
âˆ´   âˆ D = âˆ B = 72Â°
and âˆ C= âˆ A = 108Â°
âˆ´   âˆ A = 108Â°, âˆ B = 72Â°, âˆ C = 108Â° and âˆ D = 72Â°

Question 6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Solution: Let ABCD be a parallelogram such that adjacent angles âˆ A = âˆ B.
Since âˆ A + âˆ B = 180Â°
âˆ´ A = âˆ B = 180Â°/2 = 90Â°
Since, opposite angles of a parallelogram are equal.
âˆ´  âˆ A= âˆ C = 90Â°
and âˆ B= âˆ D = 90Â°
Thus,  âˆ A = 90Â°, âˆ B = 90Â°,âˆ C = 90Â° and âˆ D = 90Â°.

Question 7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.
Solution:

y + z = 70Â° â€¦(1)
In a triangle, exterior angles is equal to the sum of interior angles opposite.

âˆ´ In Î” âˆ HOP, âˆ HOP = 180Â° -(y + z)
= 180Â° â€“ 70Â°
= 110Â°
Now x = âˆ HOP    [Opposite angles of a parallelogram are equal]
âˆ´ x = 110Â°    EH || OP and PH is a transversal.
âˆ´ y = 40Â°    [Alternate angles are equal]
From (1),   40Â° + z = 70Â°
âˆ´   z = 70Â° â€“ 40Â° = 30Â°

Thus, x = 110Â°, y = 40Â° and z = 30Â°

Question 8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm.)

Solution: (i) âˆµ GUNS is a parallelogram.
âˆ´ Its opposite sides are equal.
âˆ´ GS = NU and SN = GU
or  3x = 18 and 26 = 3y â€“ 1
Now 3x = 18
â‡’   x = 18/3 = 6
3y â€“ 1 = 26
â‡’   y = 26 + 1/3 = 27/3 = 9

Thus, x = 6 cm and y = 9 cm
(ii) RUNS is a parallelogram and thus its diagonals bisect each other.
âˆ´  x + y = 16 and 7 + y = 20

i.e. y = 20 â€“ 7
or   y = 13
âˆ´  From x + y = 16, we have
x + 13 = 16

or   x = 16 â€“ 13 = 3

Thus, x = 3 cm and y = 13 cm.

Question 9. In the following figure, both RISK and CLUE are parallelogram. Find the value of x.
Solution:

RISK is a parallelogram.
âˆ´  âˆ R + âˆ K = 180Â°     [âˆµ Adjacent angles of a parallelogram are supplementary]
or  âˆ R + 120Â° = 180Â°
â‡’  âˆ R = 180Â° â€“ 120Â° = 60Â°
But  âˆ R and âˆ S are opposite angles.
âˆ´ âˆ S = 60Â°
CLUE is also a parallelogram.
âˆ´ Its opposite angles are equal.
âˆ´ âˆ E= âˆ L = 70Â°

Now, in Î” ESO, we have
âˆ E + âˆ S + x = 180Â°
âˆ´  70Â° + 60Â° + x = 180Â°
or  x = 180Â° â€“ 60Â° â€“ 70Â°

â‡’ x = 50

Question 10. Explain how this figure is a trapezium. Which of its two sides are parallel?Â°

Solution: Since, 100Â° + 80Â° = 180Â°
i.e. âˆ M and âˆ L are supplementary.
[âˆµ If interior opposite angles along the transversal are supplementary]

Question 11. Find mâˆ C in the adjoining figure if  .
Solution:

âˆµ ABCD is a trapezium in which
and BC is a transversal.
âˆ´   Interior opposite angles along BC are supplementary.
âˆ´   mâˆ B + mâˆ C = 180Â°
or  mâˆ C = 180Â° â€“ mâˆ B
âˆ´   mâˆ C = 180Â° â€“ 120Â°        [âˆµ âˆ B = 120Â°]

or mâˆ C = 60Â°

Question 12. Find the measure of âˆ P and âˆ  S if   in figure. (If you find mâˆ R, is there more than one method to find mâˆ P?)
Solution:

PQRS is a trapezium such that  and PQ is a transversal.
âˆ´  mâˆ P + mâˆ Q = 180Â°     [Interior opposite angles are supplementary]
or  mâˆ P + 130Â° = 180Â° or mâˆ P = 180Â° â€“ 130Â° = 50Â°
Again SP || RQ and RS is a transversal.

âˆ´ mâˆ S + mâ€“R = 180Â°

or  mâ€“S + 90Â° = 180Â°
âˆ´ mâˆ S = 180Â° â€“ 90Â° = 90Â°
Yes, using the angle sum property of a quadrilateral, we can find mâˆ P when mâˆ R is known.
âˆ´ mâˆ P + mâ€“Q + mâˆ R + mâˆ S = 360Â°
or  mâˆ P + 130Â° + 90Â° + 90Â° = 360Â°
or mâˆ P = 360Â° â€“ 130Â° â€“ 90Â° â€“ 90Â° = 50Â°

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## Mathematics (Maths) Class 8

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