{"id":1303,"date":"2020-10-20T03:31:15","date_gmt":"2020-10-20T03:31:15","guid":{"rendered":"http:\/\/themindpalace.in\/?p=1303"},"modified":"2021-08-26T05:32:01","modified_gmt":"2021-08-26T05:32:01","slug":"quadrilaterals","status":"publish","type":"post","link":"https:\/\/themindpalace.in\/index.php\/2020\/10\/20\/quadrilaterals\/","title":{"rendered":"QUADRILATERALS"},"content":{"rendered":"\n

Summary of quadrilaterals<\/a><\/p>\n\n\n\n

Solved exercise of quadrilaterals<\/a><\/p>\n\n\n\n

<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Summary<\/h1>\n\n\n\n

As the word \u2018Quad\u2019 means four, and lateral means sides.A quadrilateral is a plane figure that has four sides or edges, and also have four corners or vertices.AC and BD are the diagonals of the quadrilateral ABCD.<\/p>\n\n\n\n

Properties of quadrilaterals<\/span><\/strong><\/p>\n\n\n\n

Angle sum property of a quadrilateral<\/span><\/strong><\/p>\n\n\n\n

Statement: sum of angles of quadrilateral <\/strong> is 360 degrees<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

\u2234 Sum of the four angles of the quadrilateral is<\/p>\n\n\n\n

\u2234 \u2220A + \u2220B + \u2220C + \u2220D = 360\u00b0<\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Types of Quadrilaterals<\/span><\/h4>\n\n\n\n

There are many types of quadrilaterals. quadrilateral have four sides, and the sum of angles of these shapes is 360 degrees.<\/p>\n\n\n\n

1.Trapezium<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

2.Parallelogram<\/strong><\/span><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

3.Squares<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

4.Rectangle<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

5.Rhombus<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

6.Kite<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Trapezium<\/span><\/strong><\/p>\n\n\n\n

A trapezium is a quadrilateral with one pair of opposite parallel sides.<\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Properties:<\/strong> <\/span>The parallel sides are called bases. The other two non-parallel sides are called legs. \u2022If the two non-parallel sides are equal and form equal angles at one of the bases, the trapezium is an isosceles trapezium.<\/p>\n\n\n\n

Isosceles Trapezoid<\/span><\/strong><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n
  • One Pair of parallel sides (AB and DC)<\/li>
  • One pair of congruent legs (DA and CB)<\/li>
  • Base angles are congruent (M<D=M<C)<\/li>
  • Diagonals are congruent (AC and BD)<\/li>
  • Opposite angles are supplementary (their sum is 180 degree)<\/li><\/ul>\n\n\n\n

    Kite<\/span><\/strong><\/p>\n\n\n\n

    A kite is a quadrilateral that has 2 pairs of equal-length sides and these sides are adjacent to each other.<\/p>\n\n\n\n

    \"\"<\/figure><\/div>\n\n\n\n

    Properties:<\/span><\/strong> The two angles are equal where the unequal sides meet. \u2022It can be viewed as a pair of congruent triangles with a common base. \u2022It has 2 diagonals that intersect each other at right angles. \u2022The longer or main diagonal bisects the other diagonal. \u2022A kite is symmetrical about its main diagonal. \u2022The shorter diagonal divides the kite into 2 isosceles triangles.<\/p>\n\n\n\n

    Parallelogram<\/span><\/strong><\/p>\n\n\n\n

    A parallelogram is a quadrilateral with two pairs of parallel sides.<\/p>\n\n\n\n

    PERIMETER OF PARALLELOGRAM:<\/span> Perimeter of a parallelogram. Is equal to perimeter of a rectangle= 2( l + b ).<\/p>\n\n\n\n

    \"\"<\/figure><\/div>\n\n\n\n

    \u2022Opposite sides are parallel and equal in length. \u2022Opposite angles are equal in measure. \u2022Adjacent angles sum up to 180 degrees. \u2022It has 2 diagonals that bisect each other. \u2022Each diagonal divides the parallelogram into 2 congruent triangles. \u2022The two diagonals divide the parallelogram into 4 triangles of equal area. \u2022Square, rectangle, and rhombus are special types of a parallelogram.<\/p>\n\n\n\n

    \"\"<\/figure><\/div>\n\n\n\n
    • The opposite side of the parallelogram are of the same length<\/li>
    • The diagonals of a parallelogram bisect each other<\/li>
    • The opposite angles are of equal measure<\/li>
    • The opposite angles are of equal measure<\/li>
    • The sum of two adjacent angles of a parallelogram is equal to 180 degrees<\/li><\/ul>\n\n\n\n

      Exercise for class 8<\/h1>\n\n\n\n

      Exercise 4.3<\/p>\n\n\n\n

      1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      (i) AD = \u2026\u2026\u2026\u2026
      (ii) \u2220DCB = \u2026\u2026\u2026
      (iii) OC = \u2026\u2026\u2026
      (iv) \u2220DAB + \u2220CDA = \u2026\u2026..<\/p>\n\n\n\n

      Solution:<\/p>\n\n\n\n

      (i) AD = BC[Opposite sides of a parallelogram are equal]<\/p>\n\n\n\n

      (iii) OC = OA[Diagonals of a parallelogram bisect each other]<\/p>\n\n\n\n

      (iv) \u2220DAB + \u2220CDA = 180[Adjacent angles of a parallelogram are supplementary]<\/p>\n\n\n\n

      2. Consider the following parallelograms. Find the values of the unknowns x, y, z<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      Solution:(i) ABCD is a parallelogram.<\/p>\n\n\n\n

      \u2220B = \u2220D [Opposite angles of a parallelogram are equal<\/p>\n\n\n\n

      \u2220D = 100\u00b0<\/p>\n\n\n\n

      \u21d2 y = 100\u00b0<\/p>\n\n\n\n

      \u2220A + \u2220B = 180\u00b0<\/p>\n\n\n\n

       [Adjacent angles of a parallelogram are supplementary]<\/p>\n\n\n\n

      \u21d2 z + 100\u00b0 = 180\u00b0<\/p>\n\n\n\n

      \u21d2 z = 180\u00b0 \u2013 100\u00b0 = 80\u00b0<\/p>\n\n\n\n

      \u2220A = \u2220C[Opposite angles of a ||gm]<\/p>\n\n\n\n

      x = 80\u00b0<\/p>\n\n\n\n

      Hence x = 80\u00b0, y = 100\u00b0 and z = 80\u00b0<\/p>\n\n\n\n

      (ii) PQRS is a parallelogram<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      \u2220P + \u2220S = 180 [Adjacent angles of parallelogram]<\/p>\n\n\n\n

      \u21d2 x + 50\u00b0 = 180\u00b0<\/p>\n\n\n\n

      x = 180\u00b0 \u2013 50\u00b0 = 130\u00b0<\/p>\n\n\n\n

      Now, \u2220P = \u2220R[Opposite angles are equal]<\/p>\n\n\n\n

      \u21d2 x = y<\/p>\n\n\n\n

      \u21d2 y = 130\u00b0<\/p>\n\n\n\n

      Also, y = z [Alternate angles<\/p>\n\n\n\n

      z = 130\u00b0<\/p>\n\n\n\n

      Hence, x = 130\u00b0, y = 130\u00b0 and z = 130<\/p>\n\n\n\n

      (iii) ABCD is a rhombus<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      [\u2235 Diagonals intersect at 90\u00b0]<\/p>\n\n\n\n

      x = 90\u00b0<\/p>\n\n\n\n

      Now in \u2206OCB,<\/p>\n\n\n\n

      x + y + 30\u00b0 = 180\u00b0(Angle sum property)<\/p>\n\n\n\n

      \u21d2 90\u00b0 + y + 30\u00b0 = 180\u00b0<\/p>\n\n\n\n

      \u21d2 y + 120\u00b0 = 180\u00b0<\/p>\n\n\n\n

      \u21d2 y = 180\u00b0 \u2013 120\u00b0 = 60\u00b0<\/p>\n\n\n\n

      y = z (Alternate angles)<\/p>\n\n\n\n

      \u21d2 z = 60\u00b0<\/p>\n\n\n\n

      Hence, x = 90\u00b0, y = 60\u00b0 and z = 60\u00b0.<\/p>\n\n\n\n

      3. Can a quadrilateral ABCD be a parallelogram if<\/p>\n\n\n\n

      (i) \u2220D + \u2220B = 180\u00b0?
      (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?
      (iii) \u2220A = 70\u00b0 and \u2220C = 65\u00b0?<\/p>\n\n\n\n

      Solution<\/p>\n\n\n\n

      (i) For \u2220D + \u2220B = 180, quadrilateral ABCD may be a parallelogram if following conditions are also fulfilled.<\/p>\n\n\n\n

      (a) The sum of measures of adjacent angles should be 180\u00b0.<\/p>\n\n\n\n

      (b) Opposite angles should also be of same measures. So, ABCD can be but need not be a parallelogram.<\/p>\n\n\n\n

      (ii) Given: AB = DC = 8 cm, AD = 4 cm, BC = 4.4 cm<\/p>\n\n\n\n

      In a parallelogram, opposite sides are equal.<\/p>\n\n\n\n

      Here AD \u2260 BC<\/p>\n\n\n\n

      Thus, ABCD cannot be a parallelogram.<\/p>\n\n\n\n

      4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      Solution:<\/p>\n\n\n\n

      ABCD is a rough figure of a quadrilateral in which \u2220A = \u2220C but it is not a parallelogram.<\/p>\n\n\n\n

      It is a kite.<\/p>\n\n\n\n

      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.<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      Solution: Let ABCD is parallelogram such that
      \u2220B : \u2220C = 3 : 2<\/p>\n\n\n\n

      Let \u2220B = 3x\u00b0 and m\u2220C = 2x\u00b0<\/p>\n\n\n\n

      \u2220B + \u2220C = 180\u00b0 (Sum of adjacent angles = 180\u00b0)<\/p>\n\n\n\n

      3x + 2x = 180\u00b0<\/p>\n\n\n\n

      \u21d2 5x = 180\u00b0
      \u21d2 x = 36\u00b0<\/p>\n\n\n\n

      Thus, \u2220B = 3 \u00d7 36 = 108\u00b0<\/p>\n\n\n\n

      \u2220C = 2 \u00d7 36\u00b0 = 72\u00b0
      \u2220B = \u2220D = 108\u00b0
      and \u2220A = \u2220C = 72\u00b0<\/p>\n\n\n\n

      Hence, the measures of the angles of the parallelogram are 108\u00b0, 72\u00b0, 108\u00b0 and 72\u00b0.<\/p>\n\n\n\n

      7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      Solution:\u2220y = 40\u00b0 (Alternate angles)
      \u2220z + 40\u00b0 = 70\u00b0 (Exterior angle property)<\/p>\n\n\n\n

      \u21d2 \u2220z = 70\u00b0 \u2013 40\u00b0 = 30\u00b0
      z = \u2220EPH (Alternate angle)<\/p>\n\n\n\n

      In \u2206EPH
      \u2220x + 40\u00b0 + \u2220z = 180\u00b0 (Adjacent angles)<\/p>\n\n\n\n

      \u21d2 \u2220x + 40\u00b0 + 30\u00b0 = 180\u00b0
      \u21d2 \u2220x + 70\u00b0 = 180\u00b0
      \u21d2 \u2220x = 180\u00b0 \u2013 70\u00b0 = 110\u00b0<\/p>\n\n\n\n

      Hence x = 110\u00b0, y = 40\u00b0 and z = 30\u00b0.<\/p>\n\n\n\n

      8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      i)GU=SN(opposite sides of a parallelogram)<\/p>\n\n\n\n

      3y-1=26<\/p>\n\n\n\n

      =3y=26+1<\/p>\n\n\n\n

      =3y=27<\/p>\n\n\n\n

      y=27\/3=9<\/p>\n\n\n\n

      similarly, GS=UN<\/p>\n\n\n\n

      3x=18<\/p>\n\n\n\n

      x=18\/3=6<\/p>\n\n\n\n

      Hence, x=6cmand y=9cm<\/p>\n\n\n\n

      (ii) Since, the diagonals of a parallelogram bisect each other<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      Therefore, OU = OS<\/p>\n\n\n\n

      \u21d2 y + 7 = 20<\/p>\n\n\n\n

      \u21d2 y = 20 \u2013 7 = 13
      Also, ON = OR<\/p>\n\n\n\n

      \u21d2 x + y = 16
      \u21d2 x + 13 = 16<\/p>\n\n\n\n

      x = 16 \u2013 13 = 3
      Hence, x = 3 cm and y = 13 cm.<\/p>\n\n\n\n

      9. In the above figure both RISK and CLUE are parallelograms. Find the value of x.<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      Solution: Here RISK and CLUE are two parallelograms.<\/p>\n\n\n\n

      \u22201 = \u2220L = 70\u00b0 (Opposite angles of a parallelogram)<\/p>\n\n\n\n

      \u2220K + \u22202 = 180\u00b0
      Sum of adjacent angles is 180\u00b0<\/p>\n\n\n\n

      120\u00b0 + \u22202 = 180\u00b0
      \u22202 = 180\u00b0 \u2013 120\u00b0 = 60\u00b0<\/p>\n\n\n\n

      In \u2206OES,\u2220x + \u22201 + \u22202 = 180\u00b0 (Angle sum property)<\/p>\n\n\n\n

      \u21d2 \u2220x + 70\u00b0 + 60\u00b0 = 180\u00b0
      \u21d2 \u2220x + 130\u00b0 = 180\u00b0<\/p>\n\n\n\n

      \u21d2 \u2220x = 180\u00b0 \u2013 130\u00b0 = 50\u00b0
      Hence x = 50\u00b0<\/strong><\/p>\n\n\n\n

      10. Explain how this figure is a trapezium. Which of its two sides are parallel?<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      Solution:\u2220M + \u2220L = 100\u00b0 + 80\u00b0 = 180\u00b0<\/p>\n\n\n\n

      \u2220M and \u2220L are the adjacent angles, and sum of adjacent interior angles is 180\u00b0<\/p>\n\n\n\n

      KL is parallel to NM
      Hence KLMN is a trapezium.<\/p>\n\n\n\n

      11. Find \u2220C in below figure if AB || DC<\/p>\n\n\n\n

      Solution:Given that AB || DC<\/p>\n\n\n\n

      \u2220B + \u2220C = 180\u00b0 (Sum of adjacent angles of a parallelogram is 180\u00b0)<\/p>\n\n\n\n

      120\u00b0 + \u2220C = 180\u00b0<\/p>\n\n\n\n

      \u2220C = 180\u00b0 \u2013 120\u00b0 = 60\u00b0
      Hence \u2220C = 60\u00b0<\/strong><\/p>\n\n\n\n

      12. Find the measure of \u2220P and \u2220S if SP || RQ in figure, is there any other method to find \u2220P?)<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      Solution:Given that \u2220Q = 130\u00b0 and \u2220R = 90\u00b0
      SP || RQ (given)<\/p>\n\n\n\n

      \u2220P + \u2220Q = 180\u00b0 (Adjacent angles)<\/p>\n\n\n\n

      \u21d2 \u2220P + 130\u00b0 = 180\u00b0
      \u21d2 \u2220P = 180\u00b0 \u2013 130\u00b0 = 50\u00b0<\/p>\n\n\n\n

      and, \u2220S + \u2220R = 180\u00b0 (Adjacent angles)<\/p>\n\n\n\n

      \u21d2 \u2220S + 90\u00b0 = 180\u00b0
      \u21d2 \u2220S = 180\u00b0 \u2013 90\u00b0 = 90\u00b0<\/p>\n\n\n\n

      Alternate Method:<\/strong><\/p>\n\n\n\n

      \u2220Q = 130\u00b0, \u2220R = 90\u00b0 and \u2220S = 90\u00b0<\/p>\n\n\n\n

      We know that
      \u2220P + \u2220Q + \u2220R + \u2220Q = 360\u00b0 (Angle sum property of quadrilateral)<\/p>\n\n\n\n

      \u21d2 \u2220P + 130\u00b0 + 90\u00b0 + 90\u00b0 = 360\u00b0
      \u21d2 \u2220P + 310\u00b0 = 360\u00b0<\/p>\n\n\n\n

      \u21d2 \u2220P = 360\u00b0 \u2013 310\u00b0 = 50\u00b0
      Hence \u2220P = 50\u00b0
      <\/p>\n\n\n\n

      Rhombus<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      A rhombus<\/strong> is a quadrilateral with sides<\/strong> of equal length<\/strong>.<\/p>\n\n\n\n

      Since the opposite sides<\/strong> of a rhombus have the same length<\/strong>, it is also a parallelogram<\/strong>.<\/p>\n\n\n\n

      The diagonals <\/strong>of a rhombus<\/strong> are perpendicular bisectors<\/em><\/strong> of one another.<\/p>\n\n\n\n

      Properties of Special Parallelograms<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      Rectangle:<\/strong> A rectangle<\/strong> is a parallelogram<\/strong> with equal angles<\/strong> and each angle is equal to 90<\/strong>\u2218.
      Properties:<\/p>\n\n\n\n

      \u2022Opposite sides<\/strong> of a rectangle are parallel<\/strong> and equal<\/strong>.<\/p>\n\n\n\n

      \u2022The length of diagonals<\/strong> of a rectangle is equal<\/strong>.<\/p>\n\n\n\n

      All the interior angles<\/strong> of a rectangle are equal to 90<\/strong>\u2218<\/p>\n\n\n\n

      \u2022The diagonals<\/strong> of a rectangle bisect <\/strong>each other at the point of intersection.<\/p>\n\n\n\n

      Square<\/strong><\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      In a square the diagonals<\/p>\n\n\n\n

      In a square all the sides are equal<\/p>\n\n\n\n

      \u2022bisect one another<\/p>\n\n\n\n

      \u2022are of equal length<\/p>\n\n\n\n

      \u2022are perpendicular to one another<\/p>\n\n\n\n

      Example:7<\/p>\n\n\n\n

      RIce is a rhombus find x,y,z justify<\/p>\n\n\n\n

      \"\"<\/figure><\/div>\n\n\n\n

      x=OE<\/p>\n\n\n\n

      =OI(diagonals bisect<\/p>\n\n\n\n

      =5<\/p>\n\n\n\n

      y=or<\/p>\n\n\n\n

      =oc (diagonal bisect<\/p>\n\n\n\n

      =12<\/p>\n\n\n\n

      z=side of the rhombu<\/p>\n\n\n\n

      =13(all sides are equal<\/p>\n\n\n\n

      Exercise 4.4<\/strong><\/p>\n\n\n\n

      1. State whether True or False. Solution:<\/p>\n\n\n\n

      • a) All rectangles are squares. False<\/li>
      • (b) All rhombuses are parallelograms. True<\/li>
      • (c) All squares are rhombuses and also rectangles. True<\/li>
      • (d) All squares are not parallelograms. False<\/li>
      • (e) All kites are rhombuses False<\/li>
      • (f) All rhombuses are kites. True<\/li>
      • (g) All parallelograms are trapeziums. True<\/li>
      • (h) All squares are trapeziums. True<\/li><\/ul>\n\n\n\n

        <\/p>\n\n\n\n

        \"\"<\/figure><\/div>\n\n\n\n

        2. Identify all the quadrilaterals that have
        (a) four sides of equal length
        (b) four right angles<\/p>\n\n\n\n

        Solution:(a) Squares and rhombuses.
        (b) Rectangles and squares.<\/p>\n\n\n\n

        3. Explain how a square is
        (i) a quadrilateral
        (ii) a parallelogram
        (iii) a rhombus
        (iv) a rectangle<\/p>\n\n\n\n

        Solution:(i) Square is a quadrilateral because it is closed with four line segments.<\/p>\n\n\n\n

        (ii) Square is a parallelogram due to the following properties:<\/p>\n\n\n\n

        (a) Opposite sides are equal and parallel.
        (b) Opposite angles are equal.<\/p>\n\n\n\n

        (iii) Square is a rhombus because its all sides are equal and opposite sides are parallel.<\/p>\n\n\n\n

        (iv) Square is a rectangle because its opposite sides are equal and has equal diagonal.<\/p>\n\n\n\n

        5. Name the quadrilaterals whose diagonals<\/strong><\/p>\n\n\n\n

        (i) bisect each other
        (ii) are perpendicular bisectors of each other
        (iii) are equal<\/p>\n\n\n\n

        Solution:<\/p>\n\n\n\n

        (i) bisect each other<\/p>\n\n\n\n

        (i) Parallelogram, rectangle, square and rhombus<\/p>\n\n\n\n

        (ii) are perpendicular bisectors of each other<\/p>\n\n\n\n

        (ii) Square and rhombus<\/p>\n\n\n\n

        (iii) are equal<\/p>\n\n\n\n

        (iii) Rectangle and square<\/p>\n\n\n\n

        5. Explain why a rectangle is a convex quadrilateral.<\/strong><\/p>\n\n\n\n

        Solution:In a rectangle, both of its diagonal lie in its interior. Hence, it is a convex quadrilateral.<\/p>\n\n\n\n

        6. ABC is a right-angled triangle and O is the midpoint of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).<\/strong><\/p>\n\n\n\n

        \"\"<\/figure><\/div>\n\n\n\n

        Solution:<\/p>\n\n\n\n

        Since the right-angled triangle ABC makes a rectangle ABCD by the dotted lines<\/p>\n\n\n\n

        therefore OA = OB = OC = OD [Diagonals of a rectangle are equal and bisect each other]
        Hence, O is equidistant from A, B and C.<\/p>\n\n\n\n

        \"\"<\/figure>\n\n\n\n\n\n\n\n

        Class 9 exercise<\/p>\n\n\n\n

        PARALLELOGRAM:<\/strong><\/p>\n\n\n\n

        A parallelogram is a quadrilateral in which opposite sides are parallel.<\/p>\n\n\n\n

        PROPERTIES OF PARALLELOGRAM:<\/strong><\/p>\n\n\n\n

        Theorem<\/strong>1: A diagonal of a parallelogram divides it into two congruent triangles.<\/p>\n\n\n\n

        \"\"<\/figure><\/div>\n\n\n\n

        Theorem <\/strong>2:The opposite sides of a parallelogram are equal.<\/p>\n\n\n\n

        \"\"<\/figure><\/div>\n\n\n\n

        Theorem 3 :<\/strong> If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.<\/p>\n\n\n\n

        Theorem 4<\/strong>. The opposite angles of a parallelogram are equal<\/p>\n\n\n\n

        \"\"<\/figure><\/div>\n\n\n\n

        Theorem 5 :<\/strong> If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.<\/p>\n\n\n\n

        Theorem 6<\/strong>. The diagonals of a parallelogram bisect each other.<\/p>\n\n\n\n

        \"\"<\/figure><\/div>\n\n\n\n

        Theorem 7<\/strong>. The diagonals of a parallelogram bisect each other<\/p>\n\n\n\n

        Theorem 8<\/strong>. A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.<\/p>\n\n\n\n

        \"\"<\/figure><\/div>\n\n\n\n

        1. The angles of quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.
        Solution:<\/p>\n\n\n\n

        \"\"<\/figure><\/div>\n\n\n\n

        Let the angles of the quadrilateral be 3x, 5x, 9x and 13x.<\/p>\n\n\n\n

        \u2234 3x + 5x + 9x + 13x = 360\u00b0 [Angle sum property of a quadrilateral]<\/p>\n\n\n\n

        \u21d2 30x = 360\u00b0<\/p>\n\n\n\n

        \u21d2      x = 360\u00b0\/30= 12\u00b0<\/p>\n\n\n\n

        \u2234 3x = 3 x 12\u00b0 = 36\u00b0<\/p>\n\n\n\n

        5x = 5 x 12\u00b0 = 60\u00b0<\/p>\n\n\n\n

        9x = 9 x 12\u00b0 = 108\u00b0<\/p>\n\n\n\n

        13a = 13 x 12\u00b0 = 156\u00b0<\/p>\n\n\n\n

        \u21d2 The required angles of the quadrilateral are 36\u00b0, 60\u00b0, 108\u00b0 and 156\u00b0.<\/p>\n\n\n\n

        2. If the diagonals of a parallelogram are equal, then show that it is a rectangle.<\/p>\n\n\n\n

        Solution:Let ABCD is a parallelogram such that AC = BD.<\/p>\n\n\n\n

        In \u2206ABC and \u2206DCB,<\/p>\n\n\n\n

        AC = DB [Given]<\/p>\n\n\n\n

        AB = DC [Opposite sides of a parallelogram]<\/p>\n\n\n\n

        BC = CB [Common]<\/p>\n\n\n\n

        \u2234 \u2206ABC \u2245 \u2206DCB [By SSS congruency]<\/p>\n\n\n\n

        \u21d2 \u2220ABC = \u2220DCB [By C.P.C.T.] \u2026(1)<\/p>\n\n\n\n

        Now, AB || DC and BC is a transversal. [ \u2235 ABCD is a parallelogram]<\/p>\n\n\n\n

        \u2234 \u2220ABC + \u2220DCB = 180\u00b0 \u2026 (2) [Co-interior angles]<\/p>\n\n\n\n

        From (1) and (2), we have<\/p>\n\n\n\n

        \u2220ABC = \u2220DCB = 90\u00b0<\/p>\n\n\n\n

        i.e., ABCD is a parallelogram having an angle equal to 90\u00b0.<\/p>\n\n\n\n

        \u2234 ABCD is a rectangle.<\/p>\n","protected":false},"excerpt":{"rendered":"

        Summary of quadrilaterals Solved exercise of quadrilaterals Summary As the word \u2018Quad\u2019 means four, and lateral means sides.A quadrilateral is a plane figure that has four sides or edges, and also have four corners...<\/p>\n","protected":false},"author":3,"featured_media":1323,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[475,14],"tags":[161],"cp_meta_data":{"_edit_lock":["1629955922:2"],"_edit_last":["2"],"_layout":["inherit"],"_thumbnail_id":["1323"],"_oembed_95287caaddeb112cd4edfcbd8e525566":["