Mathematics · Textbook solutions

Parallel Lines

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 31 questions

Practice Set 2.1

11 q
  1. In figure 2.5, line RP ∥\parallel line MS and line DK is their transversal. ∠\angleDHP =85∘= 85^\circ. Find the measures of the following angles.
    Ex 2.1 Q1(i)
    ∠\angleRHD
  2. Ex 2.1 Q1(ii)
    ∠\anglePHG
  3. Ex 2.1 Q1(iii)
    ∠\angleHGS
  4. Ex 2.1 Q1(iv)
    ∠\angleMGK
  5. Ex 2.1 Q2
    In figure 2.6, line p∥p \parallel line qq and line ll and line mm are transversals. Measures of some angles are shown. Hence find the measures of ∠a\angle a, ∠\angleb, ∠\anglec, ∠\angled.
  6. Ex 2.1 Q3
    In figure 2.7, line l∥l \parallel line mm and line n∥n \parallel line pp. Find ∠a\angle a, ∠\angleb, ∠\anglec from the given measure of an angle.
  7. Ex 2.1 Q4
    In figure 2.8, sides of ∠\anglePQR and ∠\angleXYZ are parallel to each other. Prove that ∠\anglePQR ≅∠\cong \angleXYZ.
  8. In figure 2.9, line AB ∥\parallel line CD and line PQ is a transversal. The measure of one of the angles is given. Hence find the measures of the following angles.
    Ex 2.1 Q5(i)
    ∠\angleART
  9. Ex 2.1 Q5(ii)
    ∠\angleCTQ
  10. Ex 2.1 Q5(iii)
    ∠\angleDTQ
  11. Ex 2.1 Q5(iv)
    ∠\anglePRB

Practice Set 2.2

6 q
  1. Ex 2.2 Q1
    In figure 2.18, y=108∘y = 108^\circ and x=71∘x = 71^\circ. Are the lines mm and nn parallel ? Justify ?
  2. Ex 2.2 Q2
    In figure 2.19, if ∠a≅∠\angle a \cong \angleb then prove that line l∥l \parallel line mm.
  3. Ex 2.2 Q3
    In figure 2.20, if ∠a≅∠\angle a \cong \angleb and ∠x≅∠y\angle x \cong \angle y then prove that line l∥l \parallel line nn.
  4. Ex 2.2 Q4
    In figure 2.21, if ray BA ∥\parallel ray DE, ∠\angleC =50∘= 50^\circ and ∠\angleD =100∘= 100^\circ. Find the measure of ∠\angleABC. (Hint : Draw a line passing through point C and parallel to line AB.)
  5. Ex 2.2 Q5
    In figure 2.22, ray AE ∥\parallel ray BD, ray AF is the bisector of ∠\angleEAB and ray BC is the bisector of ∠\angleABD. Prove that line AF ∥\parallel line BC.
  6. Ex 2.2 Q6
    A transversal EF of line AB and line CD intersects the lines at point P and Q respectively. Ray PR and ray QS are parallel and bisectors of ∠\angleBPQ and ∠\anglePQC respectively. Prove that line AB ∥\parallel line CD.

Problem Set 2

14 q
  1. Select the correct alternative and fill in the blanks in the following statements.
    Prob Q1(i)
    If a transversal intersects two parallel lines then the sum of interior angles on the same side of the transversal is ............
    1. A.
      0∘0^\circ
    2. B.
      90∘90^\circ
    3. C.
      180∘180^\circ
    4. D.
      360∘360^\circ
  2. Prob Q1(ii)
    The number of angles formed by a transversal of two lines is ............
    1. A.
      2
    2. B.
      4
    3. C.
      8
    4. D.
      16
  3. Prob Q1(iii)
    A transversal intersects two parallel lines. If the measure of one of the angles is 40∘40^\circ then the measure of its corresponding angle is ............
    1. A.
      40∘40^\circ
    2. B.
      140∘140^\circ
    3. C.
      50∘50^\circ
    4. D.
      180∘180^\circ
  4. Prob Q1(iv)
    In Δ\DeltaABC, ∠\angleA =76∘= 76^\circ, ∠\angleB =48∘= 48^\circ, ∴∠\therefore \angleC == ..............
    1. A.
      66∘66^\circ
    2. B.
      56∘56^\circ
    3. C.
      124∘124^\circ
    4. D.
      28∘28^\circ
  5. Prob Q1(v)
    Two parallel lines are intersected by a transversal. If the measure of one of the alternate interior angles is 75∘75^\circ then the measure of the other angle is ............
    1. A.
      105∘105^\circ
    2. B.
      15∘15^\circ
    3. C.
      75∘75^\circ
    4. D.
      45∘45^\circ
  6. Ray PQ and ray PR are perpendicular to each other. Points B and A are in the interior and exterior of ∠\angleQPR respectively. Ray PB and ray PA are perpendicular to each other. Draw a figure showing all these rays and write -
    Prob Q2(i)
    A pair of complementary angles.
  7. Prob Q2(ii)
    A pair of supplementary angles.
  8. Prob Q2(iii)
    A pair of congruent angles.
  9. Prob Q3
    Prove that, if a line is perpendicular to one of the two parallel lines, then it is perpendicular to the other line also.
  10. Prob Q4
    In figure 2.24, measures of some angles are shown. Using the measures find the measures of ∠x\angle x and ∠y\angle y and hence show that line l∥l \parallel line mm.
  11. Prob Q5
    Line AB ∥\parallel line CD ∥\parallel line EF and line QP is their transversal. If y:z=3:7y : z = 3 : 7 then find the measure of ∠x\angle x. (See figure 2.25.)
  12. Prob Q6
    In figure 2.26, if line q∥q \parallel line rr, line pp is their transversal and if a=80∘a = 80^\circ find the values of ff and gg.
  13. Prob Q7
    In figure 2.27, if line AB ∥\parallel line CF and line BC ∥\parallel line ED then prove that ∠\angleABC =∠= \angleFDE.
  14. Prob Q8
    In figure 2.28, line PS is a transversal of parallel line AB and line CD. If ray QX, ray QY, ray RX, ray RY are angle bisectors, then prove that □\squareQXRY is a rectangle.