Mathematics · Textbook solutions

Quadrilaterals

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 49 questions

Properties of a Parallelogram

2 q

Solved Examples

Worked · 2
  1. 5.1 SolvedEx.1
    □\squarePQRS is a parallelogram. PQ = 3.5, PS = 5.3, ∠Q=50∘\angle Q = 50^{\circ} then find the lengths of remaining sides and measures of remaining angles.
  2. 5.1 SolvedEx.2
    □\squareABCD is a parallelogram. If ∠A=(4x+13)∘\angle A = (4x + 13)^{\circ} and ∠D=(5x−22)∘\angle D = (5x - 22)^{\circ} then find the measures of ∠B\angle B and ∠C\angle C.

Practice Set 5.1

7 q
  1. Ex 5.1 Q1
    Diagonals of a parallelogram WXYZ intersect each other at point O. If ∠XYZ=135∘\angle XYZ = 135^{\circ} then what is the measure of ∠XWZ\angle XWZ and ∠YZW\angle YZW ? If l(OY)=5l(OY) = 5 cm then l(WY)=?l(WY) = ?
  2. Ex 5.1 Q2
    In a parallelogram ABCD, If ∠A=(3x+12)∘\angle A = (3x + 12)^{\circ}, ∠B=(2x−32)∘\angle B = (2x - 32)^{\circ} then find the value of xx and then find the measures of ∠C\angle C and ∠D\angle D.
  3. Ex 5.1 Q3
    Perimeter of a parallelogram is 150 cm. One of its sides is greater than the other side by 25 cm. Find the lengths of all sides.
  4. Ex 5.1 Q4
    If the ratio of measures of two adjacent angles of a parallelogram is 1 : 2, find the measures of all angles of the parallelogram.
  5. Ex 5.1 Q5*
    Diagonals of a parallelogram intersect each other at point O. If AO = 5, BO = 12 and AB = 13 then show that □\squareABCD is a rhombus.
  6. Ex 5.1 Q6
    In the figure 5.12, □\squarePQRS and □\squareABCR are two parallelograms. If ∠P=110∘\angle P = 110^{\circ} then find the measures of all angles of □\squareABCR. [In Fig. 5.12 the vertices of □\squarePQRS are P (top left), Q (top right), R (bottom right) and S (bottom left), and the angle of 110∘110^{\circ} is marked at P between side PS and side PQ. Point A lies on side SR, point C lies on side QR, and B lies inside the parallelogram, so that □\squareABCR shares the vertex R with □\squarePQRS, with side AR along SR and side CR along QR.]
  7. Ex 5.1 Q7
    In figure 5.13 □\squareABCD is a parallelogram. Point E is on the ray AB such that BE = AB then prove that line ED bisects seg BC at point F. [In Fig. 5.13 the vertices of □\squareABCD are A (top left), B (top right), C (bottom right) and D (bottom left). E lies on ray AB beyond B, with seg AB and seg BE marked congruent, and the line ED crosses seg BC at the point F.]

Tests for a Parallelogram

3 q

Solved Examples

Worked · 3
  1. 5.2 SolvedEx.1
    □\squarePQRS is parallelogram. M is the midpoint of side PQ and N is the mid point of side RS. Prove that □\squarePMNS and □\squareMQRN are parallelograms.
  2. 5.2 SolvedEx.2
    Points D and E are the midpoints of side AB and side AC of △\triangleABC respectively. Point F is on ray ED such that ED = DF. Prove that □\squareAFBE is a parallelogram. For this example write 'given' and 'to prove' and complete the proof given below.
  3. 5.2 SolvedEx.3
    Prove that every rhombus is a parallelogram.

Practice Set 5.2

5 q
  1. Ex 5.2 Q1
    In figure 5.22, □\squareABCD is a parallelogram, P and Q are midpoints of side AB and DC respectively, then prove □\squareAPCQ is a parallelogram. (In the figure the vertices run A at the top left, D at the top right, C at the bottom right and B at the bottom left, so side AB is the left side and side DC the right side; P lies on side AB and Q on side DC, and seg AQ and seg PC are drawn, forming the quadrilateral A-P-C-Q.)
  2. Ex 5.2 Q2
    Using opposite angles test for parallelogram, prove that every rectangle is a parallelogram.
  3. Ex 5.2 Q3
    In figure 5.23, G is the point of concurrence of medians of △\triangleDEF. Take point H on ray DG such that D-G-H and DG = GH, then prove that □\squareGEHF is a parallelogram. (In the figure D is at the top with E at the lower left and F at the lower right; the medians of △\triangleDEF meet at G, and ray DG is produced through the midpoint of seg EF to the point H below seg EF, with seg GE, seg EH, seg HF and seg FG drawn.)
  4. Ex 5.2 Q4
    Prove that quadrilateral formed by the intersection of angle bisectors of all angles of a parallelogram is a rectangle. (Figure 5.24) (In the figure the parallelogram is ABCD with A at the top left, B at the top right, C at the bottom right and D at the bottom left; the bisectors of ∠\angleA, ∠\angleB, ∠\angleC and ∠\angleD are drawn as dashed rays and meet in pairs at the four points P, Q, R and S, which form the quadrilateral □\squarePQRS inside the parallelogram.)
  5. Ex 5.2 Q5
    In figure 5.25, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that □\squarePQRS is a parallelogram. (In the figure the parallelogram is ABCD with A at the top left, B at the top right, C at the bottom right and D at the bottom left; P lies on side AB, Q on side BC, R on side CD and S on side DA, the equal parts AP, BQ, CR and DS are marked with single ticks, and seg PQ, seg QR, seg RS and seg SP are drawn.)

Practice Set 5.3

10 q
  1. Ex 5.3 Q1
    Diagonals of a rectangle ABCD intersect at point O. If AC=8AC = 8 cm then find the length of BO and if ∠\angleCAD =35∘= 35^\circ then find the measure of ∠\angleACB.
  2. Ex 5.3 Q2
    In a rhombus PQRS if PQ=7.5PQ = 7.5 then find the length of QR. If ∠\angleQPS =75∘= 75^\circ then find the measure of ∠\anglePQR and ∠\angleSRQ.
  3. Ex 5.3 Q3
    Diagonals of a square IJKL intersects at point M, Find the measures of ∠\angleIMJ, ∠\angleJIK and ∠\angleLJK .
  4. Ex 5.3 Q4
    Diagonals of a rhombus are 20 cm and 21 cm respectively, then find the side of rhombus and its perimeter.
  5. State with reasons whether the following statements are 'true' or 'false'.
    Ex 5.3 Q5(i)
    Every parallelogram is a rhombus.
  6. Ex 5.3 Q5(ii)
    Every rhombus is a rectangle.
  7. Ex 5.3 Q5(iii)
    Every rectangle is a parallelogram.
  8. Ex 5.3 Q5(iv)
    Every squre is a rectangle.
  9. Ex 5.3 Q5(v)
    Every square is a rhombus.
  10. Ex 5.3 Q5(vi)
    Every parallelogram is a rectangle.

Rhombus, Rectangle and Square

2 q

Solved Examples

Worked · 2
  1. 5.4 SolvedEx.1
    Measures of angles of □\squareABCD are in the ratio 4 : 5 : 7 : 8. Show that □\squareABCD is a trapezium.
  2. 5.4 SolvedEx.2
    In □\squarePQRS, side PS ∥\parallel side QR and side PQ ≅\cong side SR, side QR >> side PS then prove that ∠\anglePQR ≅\cong ∠\angleSRQ

Practice Set 5.4

3 q
  1. Ex 5.4 Q1
    In □\squareIJKL, side IJ ∥\parallel side KL ∠\angleI =108∘= 108^\circ ∠\angleK =53∘= 53^\circ then find the measures of ∠\angleJ and ∠\angleL.
  2. Ex 5.4 Q2
    In □\squareABCD, side BC ∥\parallel side AD, side AB ≅\cong side DC If ∠\angleA =72∘= 72^\circ then find the measures of ∠\angleB, and ∠\angleD.
  3. Ex 5.4 Q3
    In □\squareABCD, side BC << side AD (Figure 5.32) side BC ∥\parallel side AD and if side BA ≅\cong side CD then prove that ∠\angleABC ≅\cong ∠\angleDCB.

Trapezium and the Midpoint Theorem

2 q

Solved Examples

Worked · 2
  1. 5.5 SolvedEx.1
    Points E and F are mid points of seg AB and seg AC of △\triangleABC respectively. If EF = 5.6 then find the length of BC.
  2. 5.5 SolvedEx.2
    Prove that the quadrilateral formed by joining the midpoints of sides of a quadrilateral in order is a parallelogram.

Practice Set 5.5

4 q
  1. Ex 5.5 Q1
    In figure 5.38, points X, Y, Z are the midpoints of side AB, side BC and side AC of △\triangleABC respectively. AB = 5 cm, AC = 9 cm and BC = 11 cm. Find the length of XY, YZ, XZ.
  2. Ex 5.5 Q2
    In figure 5.39, □\squarePQRS and □\squareMNRL are rectangles. If point M is the midpoint of side PR then prove that, (i) SL = LR, (ii) LN=12SQLN = \frac{1}{2} SQ.
  3. Ex 5.5 Q3
    In figure 5.40, △\triangleABC is an equilateral traingle. Points F,D and E are midpoints of side AB, side BC, side AC respectively. Show that △\triangleFED is an equilateral traingle.
  4. Ex 5.5 Q4
    In figure 5.41, seg PD is a median of △\trianglePQR. Point T is the mid point of seg PD. Produced QT intersects PR at M. Show that PMPR=13\frac{PM}{PR} = \frac{1}{3}. [Hint : draw DN ∥\parallel QM.]

Problem Set 5

11 q
  1. Choose the correct alternative answer and fill in the blanks.
    Prob Q1(i)
    If all pairs of adjacent sides of a quadrilateral are congruent then it is called ....
    1. A.
      rectangle
    2. B.
      parallelogram
    3. C.
      trapezium
    4. D.
      rhombus
  2. Prob Q1(ii)
    If the diagonal of a square is 12212\sqrt{2} cm then the perimeter of square is ......
    1. A.
      24 cm
    2. B.
      24224\sqrt{2} cm
    3. C.
      48 cm
    4. D.
      48248\sqrt{2} cm
  3. Prob Q1(iii)
    If opposite angles of a rhombus are (2x)∘(2x)^{\circ} and (3x−40)∘(3x - 40)^{\circ} then value of xx is ...
    1. A.
      100∘100^{\circ}
    2. B.
      80∘80^{\circ}
    3. C.
      160∘160^{\circ}
    4. D.
      40∘40^{\circ}
  4. Prob Q2
    Adjacent sides of a rectangle are 7 cm and 24 cm. Find the length of its diagonal.
  5. Prob Q3
    If diagonal of a square is 13 cm then find its side.
  6. Prob Q4
    Ratio of two adjacent sides of a parallelogram is 3:43 : 4, and its perimeter is 112 cm. Find the length of its each side.
  7. Prob Q5
    Diagonals PR and QS of a rhombus PQRS are 20 cm and 48 cm respectively. Find the length of side PQ.
  8. Prob Q6
    Diagonals of a rectangle PQRS are intersecting in point M. If ∠\angleQMR =50∘= 50^{\circ} then find the measure of ∠\angleMPS.
  9. Prob Q7
    In the adjacent Figure 5.42, if seg AB ∥\parallel seg PQ , seg AB ≅\cong seg PQ, seg AC ∥\parallel seg PR, seg AC ≅\cong seg PR then prove that, seg BC ∥\parallel seg QR and seg BC ≅\cong seg QR.
  10. Prob Q8*
    In the Figure 5.43, □\squareABCD is a trapezium. AB ∥\parallel DC. Points P and Q are midpoints of seg AD and seg BC respectively. Then prove that, PQ ∥\parallel AB and PQ=12(AB+DC)PQ = \frac{1}{2}(AB + DC).
  11. Prob Q9
    In the adjacent figure 5.44, □\squareABCD is a trapezium. AB ∥\parallel DC. Points M and N are midpoints of diagonal AC and DB respectively then prove that MN ∥\parallel AB.