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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) m ∠DAB + m ∠CDA = ...... parallelogram

Solution :-
(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)  m ∠DAB + m ∠CDA = = 180 [Adjacent angles in a parallelogram are supplementary]

    

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

Solution :-
(i) ∠ B + ∠ C = 180° [Adjacent angles in a parallelogram are supplementary]
    100° + x = 180°
    And Also,
    x = 180° - 100° = 80°
    z = x = 80° [Since opposite angles of a parallelogram are equal]
    y = 100° [Since opposite angles of a parallelogram are equal]
(ii) x + 50° = 180° [Adjacent angles in a //gm are supplementary]
	x = 180° - 50° = 130°
	z = x = 130° [Corresponding angles]
(iii) x = 90° [Vertically opposite angles]
    y + x + 30° = 180° [Angle sum property of triangle] 
    y + 90° + 30° = 180°
    y + 120° = 180°
    y = 180° - 120° 
    y = 60°
    z = y = 60° [Alternate angles]
        
(iv) z = 80° [Corresponding angles]
    x + 80° = 180° [Adjacent angles in a //gm are supplementary]
          x = 180° - 80°
          x = 100°
    y = x = 100° [opposite angles are equal in a //gm]
        
(v) y = 112° [opposite angles]
    x + y + 40° = 180° [Angle sum property of triangle] 
    x + 112° + 40° = 180°
    x + 152° = 180°
    x = 180° - 152° 
    x = 28°
    z = x = 28° [Alternate angles]
    

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

Solution :-
(i)It can be , but not always as you need to look for other criteria as well.

(ii) In a parallelogram opposite sides are always equal, here AD BC, so its not a parallelogram.

(iii) Here opposite angles are not equal, so 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 measure.

Solution :-
  ABCD is a quadrilateral in which angles ∠ A = ∠ C = 110°.
  Therefore, it could be a kite.
quadrilateral
    

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 :-
  parallelogram
  Let two adjacent angles be  3x  and 2x.
  Since the adjacent angles in a parallelogram are supplementary.
  3x + 2x = 180°		
       5x = 180°
        x = 180°/5 
        x = 36°
  one angle = 3x = 3 x 36° = 108°
  second angle = 2x = 2 x 36° = 72°
    

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

Solution :-
  Let each adjacent angle be x.
  Since the adjacent angles in a parallelogram are supplementary.
  x + x = 180°
     2x = 180°
      x = 180°/2
      x = 90°
  So, each adjacent angle is 90°.
    

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. parallelogram

Solution :-
  Angle opposite to y = 180° - 70° = 110°
  Hence, y = 40° [alternate angle of ∠ PHE]
  x = 180° - (110° + 40°) = 30°, [triangle’s angle sum]
  z = 30° [Alternate angle of a transversal]
    

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

Solution :-
(i) As opposite sides are equal in a parallelogram.
  so, In //gm GUNS
  3y - 1 = 26
  3y = 26 + 1
  3y = 27
   y = 27/3
   y = 9 cm
  similarly, 
  3x = 18
   x = 18/3
   x = 6 cm
(ii) As you know diagonals bisect each other in a parallelogram.
  so, In //gm RUNS
   y + 7 = 20   [Diagonals of //gm bisects each other]
       y = 20 - 7
       y = 13 cm
  similarly.
  x + y = 16
  x + 13 = 16
  x = 16 - 13
  x = 3 cm 
    

Question-9 :-  parallelogram
In the above figure both RISK and CLUE are parallelograms. Find the value of x.

Solution :-
  In //gm RISK,
  ∠ RIS = ∠ K = 120°   [opposite angles of //gm are equal]
  ∠ m + 120° = 180°      [Linear Pair]
  ∠ m = 180° - 120°
  ∠ m = 60°

  Now, ∠ ECI = ∠ L = 70°  [Corresponding Angles]
        m + n + ∠ ECI = 180° [Angle sum property of triangle]
        60° + n + 70° = 180°
        130° + n = 180°
        n = 180° - 130°
        n = 50°
  Also, x = n = 50° [Vertically opposite angles]
    

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

Solution :-
  Here, ∠M + ∠L = 100° + 80° = 180°  [sum of interior opposite angles]
  NM and KL are parallel.
  So, KLMN is a trapezium.
    

Question-11 :-  Find m∠C in Figure if AB||DC . quadrilateral

Solution :-
  Here, ∠B + ∠C = 180°  [AB||DC]
  120° + m∠C = 180°
  m∠C = 180° - 120°
  m∠C = 60°
    

Question-12 :-  Find the measure of ∠P and ∠S if SP||RQ in Figure. (If you find m∠R, is there more than one method to find m∠P?) parallel

Solution :-
  Here, ∠P + ∠Q = 180°  [sum of co-interior angles]
        ∠P + 130° = 180°
        ∠P = 180° - 130°
        ∠P = 50°
        ∠R = 90°
        ∠S + 90° = 180°  [Given]
        ∠S = 180° - 90°
        ∠S = 90°
    
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