Understanding Quadrilaterals
Exercise 3.3
Question 1:
Given a parallelogram ABCD. Complete each statement along with the definition or property used.
(i) AD = ____________________
(ii) DCB = ____________________
(iii) OC = ____________________
(iv) mDAB + mCDA = ____________________
Answer 1:
(i) AD = BC [Since opposite sides of a parallelogram are equal]
(ii) DCB = DAB [Since opposite angles of a parallelogram are equal]
(iii) OC = OA [Since diagonals of a parallelogram bisect each other]
(iv) mDAB + mCDA = 180^{o }[Adjacent angles in a parallelogram are supplementary]
Question 2:
Consider the following parallelograms. Find the values of the unknowns x, y, ∠.
Note: For getting correct answer, read 3^{o}= 30^{o} in figure (iii)
(i) B + C = 180^{o} [Adjacent angles in a parallelogram are supplementary]
100^{o}+ x =180^{o}
x =180^{o}100^{o} = 80^{o}
and ∠= x = 80^{o} [Since opposite angles of a parallelogram are equal]
also y =100^{o }[Since opposite angles of a parallelogram are equal]
(ii) x +50^{o} =180^{o} [Adjacent angles in a gm are supplementary]
x =180^{o} 50^{o} =130^{o}
∠= x =130^{o} [Corresponding angles]
(iii) x = 90^{o} [Vertically opposite angles]
y + x +30^{o} =180^{o} [Angle sum property of a triangle]
[Alternate angles]
(iv) ∠= 80^{o} Corresponding angles]
x +80^{o} = 180^{o} [Adjacent angles in a gm are supplementary]
x =180^{o} 80^{o} = 100^{o }
and y = 80^{o }[Opposite angles are equal in a gm]
(v) y =112^{o} [Opposite angles are equal in a gm]
40^{o} + y + x =180^{o} [Angle sum property of a triangle]
40^{o}+112^{o}+ x =180^{o}
152^{o}+ x =180^{o}
x =180^{o}^{ }152^{o} = 28^{o}
and ∠= x = 28^{o} [Alternate angles]
Question 3:
Can a quadrilateral ABCD be a parallelogram, if:
(i) D + B = 180^{o}
(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
(iii) A = 70^{o} and C = 65^{o}?
Answer 3:
(i) D + B = 180^{o}
It can be, but here, it needs not to be.
(ii) No, in this case because one pair of opposite sides are equal and another pair of opposite sides are unequal. So, it is not a parallelogram.
(iii) No. A C.
Since opposite angles are equal in parallelogram and here opposite angles are not equal in quadrilateral ABCD. Therefore it is not a parallelogram.
Question 4:
Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measures.
Answer 4:
ABCD is a quadrilateral in which angles A = C = 110^{o}.
Therefore, it could be a kite.
Question 5:
The measure 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 5:
Let two adjacent angles be 3x and 2 . x
Since the adjacent angles in a parallelogram are supplementary.
One angle = 3x = 3x36^{o} =108^{o}
and another angle = 2x = 2x36^{o} = 72^{o}
Question 6:
Two adjacent angles of a parallelogram have equal measure. Find the measure of the angles of the parallelogram.
Answer 6:
Let each adjacent angle be x.
Since the adjacent angles in a parallelogram are supplementary.
x + x =180^{o}
2x =180^{o}
Question 7:
The adjacent figure HOPW is a parallelogram. Find the angle measures x, y and ∠. State the properties you use to find them.
Answer 7:
Here HOP + 70^{o} =180^{o}[Angles of linear pair]
HOP = 180^{o }70^{o} = 110^{o}
and E = HOP [Opposite angles of a ^{gm} are equal]
x =110^{o}
∠ PHE = ∠HPO [Alternate angles]
∴ y = 40^{0}
Now ∠ EHO = ∠O = 70° [Corresponding angles]
=> 40° + ∠ = 70°
=> ∠ = 70° 40° = 30°
Hence,
x = 110° , y = 40° and ∠ =30°
Question 8:
The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)
Answer 8:
(i) In parallelogram GUNS
[ Opposite sides of parallelogram are equal]
[ Opposite sides of parallelogram are equal]
Hence, x = 6cm, and y = 9 cm
(ii) In parallelogram RUNS
y + 7 = 20 [ Diagonals of ^{gm} bisects each other]
Question 9:
In the figure, both RISK and CLUE are parallelograms. Find the value of x.
Answer 9:
In Parallelogram RISK,
∠RIS = ∠K = 120^{0}
[Opposite angles of A ^{gm} bisects each other]
∠m +120° = 180° [Linear pair]
=> ∠m = 180°120° = 60°
and
=> ∠ ECI = ∠ L = 70° [Corresponding angles]
=> m + n + ∠ECl = 180° [Angle sum property of a triangle]
60°+n+ 70° = 180°
=> 130°+n = 180°
=> n= 180°130° = 50°
also x = n = 50° [Vertically opposite angles]
Question 10:
Explain how this figure is a trape∠ium. Which is its two sides are parallel?
Answer 10:
Here, ∠M + ∠L = 100° + 80 ° = 180° [Sum of interior opposite angles is 180°]
NM and KL are parallel.
Hence, KLMN is a trapezium
Question 11:
Answer 11:
Here, ∠B + ∠C = 180^{0}
Question 12:
Find the measure of ∠P and ∠S if SP  RQ in given figure.
(If you find m∠R is there more than one method to find m∠P)
Answer 12:
Here, ∠P + ∠Q = 180° [Sum of cointerior angles is 180° ]
=> ∠P + 130° = 180°
∠P = 180°130°
=> ∠P = 50°
∴∠R = 90^{0}
∠S + 90° = 180° [Given]
∠S = 180°90°
=> ∠s = 90°
Yes, one more method is there to find ∠P.
∠S+∠R+∠Q + ∠P = 360° [Angle sum property of quadrilateral]
90°+90° +130°+∠P = 360°
=> 310° + ∠P = 360°
=> ∠P = 360°310°
=> ∠P = 50°
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