6) Lines & Angles - Questions

1. In the given figure, 2b - a = 650 and ÐBOC = 900, findthe measure of ÐAOB,ÐAOD andÐCOD.

2. In fig. ray OE bisect ÐAOB and OF is a ray opposite toOE. Show that ÐFOB =ÐFOA.

3. In fig. ray OE bisects ÐAOC and OF bisects ÐCOB andOE OF. Show that A, O, B are collinear.

4. In fig, three lines p, q and r are concurrent at O. If a =500 and b = 900, find c, d, e and f.

5. AB, CD and EF are three concurrent lines passingthrough the point O such that OF bisects ÐBOD. If ÐBOF= 350 find ÐBOC andÐAOD.

6. Prove that the bisectors of a pair of vertically oppositeangles are in the same straight line.

ORAB and CD are two intersecting lines. OP and OQ arerespectively bisectors of ÐBOD and ÐAOC. Show thatOP and OQ are opposite rays.

7. One of the four angles formed by two intersecting linesis a right angle. Show that the other three angles willalso be right angles.

8. In fig. given that AB||CD.(i) If Ð1 = (120 - x)0 and Ð5 = 5x0, find the measures ofÐ1 and Ð5.(ii) If Ð2 = (3x - 10)0 and Ð8 = (5x - 30)0, find themeasures of Ð2 and Ð8.

9. In figure if BA || DF, AD || FG, ÐBAC = 650 and ÐACB =550, then findÐFGH.

10. In figure, AB, CD and EF are three parallel lines, if 4y = 5xand z = y - 30, findÐq.

11. In fig, if p is transversal to lines q and r, q||r and angles1 and 2 are in the ratio 3 : 2, find all the angles.

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12. In fig. p is a transversal to lines m and n, Ð1 = 600 and

Ð2 = of a right angle. Prove that m||n.

13. In fig, n||m and p||q. If Ð1 = 750, prove that Ð2 = Ð1 +

of a right angle.

14. Find the value of x, if(i) In fig, AB || CD

(ii) In fig, AB||CD and BC||DE

15. In fig, AB||CF and BC||ED. Prove that ÐABC =ÐFDE.

16. If fig. if I, m and n are parallel lines, p is a transversal andÐ1 = 600, then find Ð2.

17. In fig, if AB||CD then find the value of(i) x (ii) ÐGHS (iii) ÐPRG

(iv) ÐCHF(v) ÐRSD

18. If fig AB || DC and AD||BC. Prove that ÐDAB = ÐDCB.


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19. In fig AB||CD. DetermineÐa.

20. One of the angles of a triangle is 650. Find the remainingtwo angles, if their deference is 250.

21. Prove that is one angle of a triangle is equal to the sumof the other two angles, the triangle is right angled.

22. Side BC and a DABC is produced is both the directions.Prove that the sum of the two exterior angles so formedis greater than 1800.

23. The side EF, FD and DE of a triangle DEF are produced inorder forming three exterior angles DFP, EDQ and FERrespectively. Prove thatÐDFP + ÐEDQ + ÐFER = 3600

24. In DABC, ÐB = 450, ÐC = 55.0 and bisector of ÐA meetsBC at a point D. Find ÐADB andÐADC.

25. Prove that if two parallel lines are intersected bytransversal, then the bisectors of the interior angles onthe same side of the transversal intersect at right angles.

26. If two angles of a triangle are equal and complementary,what kind of triangle is it ?

27. If fig, show thatÐA +ÐB +ÐC + ÐD + ÐE + ÐF = 3600

28. The side BC of a triangle ABC is produced to ray BC suchthat D is on ray BC. The bisector of ÐA meets BC is L.Prove that ÐABC +ÐACD = 2ÐALC.

29. Two angles of a triangle are equal and the third angle isgreater than each of these angles by 300. Find all theangles of the triangle.

30. The side BC of a triangle ABC has bee produced bothways to D and E. If ÐABD = 1250 and ÐACE = 1300, thenfind ÐBAC.

31. In fig. write all pairs of adjacent angles and all the linearpairs.

32. How many pairs of adjacent angles are formed whentwo lines intersect in a point ?

33. How many pairs of adjacent angles, in all, can you namein fig.

34. Find the measure of the complementary angle of each ofthe following angles :(i) 200 (ii) 770 (iii)900

35. Find the measure of the supplementary angle of each ofthe following angles :(i) 1320 (ii) 540 (iii)1380

36. If an angle is 280 les than its complement, find itsmeasure.

37. If an angle is 300 more than one half of its complementfind the measure of the angle.

38. Two supplementary angles are in the ratio 4 : 5. Find theangles.

39. Two supplementary angle differ by 480. Find the angles.40. If the angle (2x - 10)0 and (x - 5)0 are complementary

angles, find x .41. If the complement of an angle is equal to the

supplement of the thrice of it, find the measure of theangle.


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42. OA and OB are opposite rays(i) If x = 750, what is the value of y ?(ii) If y = 1100, what is the value of x ?

43. ÐPOR and ÐQOR form a linear pair. If a - b = 80, find thevalues of a and b.

44. Find x, further findÐBOC,ÐCOD andÐAOD

45. Determine the value of x.

46. In fig. find the values of x, y and z.

47. In fig. find the value of x.

48. In figure, if ÐBOC = 7x + 200 and ÐCOA = 3x, then findthe value of x for which AOB becomes a straight line

49. In figure, findÐCOD when ÐAOC + ÐBOD = 1000.

50. In figure, x : y : z = = 5 : 4 : 6. If XOY is a straight line findthe values of x, y and z.

51. In the given figure, AB is a mirror, PO is the incident rayand OR, the reflected ray. If ÐPOR = 1120 findÐPOA.


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52. In fig. if x = y and x = b, prove that r|| n.

53. If p, m, n are three lines such that p||m and n p, provethat n m.

54. A transversal intersects two given lines in such a waythat the interior angles on the same side of thetransversal are equal. Is it always true that the givenlines are parallel ? If not, state the condition(s) underwhich the two lines will be parallel.

55. If the angles of a triangle are in the ratio 2 : 3 : 4, findthe three angles.

56. The sides AB and AC of a DABC are produced to P and Qrespectively. The bisectors of ÐPBC and ÐQCB intersect

at O. Prove thatÐBOC = 900- ÐA.57. In figure, PT ^ QR and PS bisects ÐQPR.

If ÐQ = 650 and ÐR = 330, find ÐTPS.

58. In figure, ABCD and BPQ are lines. BP = BC and DQ || CP.Prove that(i) CP = CD (ii) DP bisectsÐCDQ.

59. ABCDE is a regular pentagon. Find each angle of DBDE.

60. In figure ÐQ >ÐR and M is a point QR such that PM isthe bisector ofÐQPR. If the perpendicular from P on QRmeets QR at N, then prove that ÐMPN = (ÐQ - ÐR).

61. In figure side QR of ÐPQR has been produced to S, if ÐP: ÐQ : ÐR = 3 : 2 : 1 and RT ^ PR, findÐTRS.

62. If the angles of a triangle are in the ratio 2 : 3 : 4, findthe three angles.

63. In figure the bisectors ofÐABC and ÐBCA, intersect

each other at point O. Prove thatÐBOC = 900 + ÐA

64. In figure, OE bisects ÐAOC, OF bisects ÐCOB and OE^ OF. Show that A, O, B are collinear.

65. In figure, AB || CD and EF || DQ. Determine ÐPDQ,ÐAED and ÐDEF.


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66. In figure, m || n and p || q. If Ð1 = 750, prove that Ð2 =

Ð1 + of a right angle.

67. Prove that two angles which have their arms parallel areeither equal or supplementary.

68. In figure, AB || DC. If x = y and y = z, find thevalues of x, y, z.

69. Two lines AB and CD intersect at O. If ÐAOC + ÐCOB +ÐBOD = 2700, find the measures of ÐAOC, ÐCOB,ÐBOD, ÐDOA.

70. If the bisectors of angles ÐABC and ÐACB of a triangleABC meet at a point O, then prove that ÐBOC = 900 +

A.71. If two parallel lines are intersected by a transversal,

prove that the bisectors of the two pairs of interiorangles enclose a rectangle.

72. The angles of a triangle are arranged in ascending orderof magnitude. If the difference between twoconsecutive angles is 10º, find all the three angles.

73. In a DABC, ÐABC = ÐACB and the two bisectors of ÐABCand ÐACB intersect at O such that ÐBOC = 1200. ShowthatÐA = ÐB =ÐC = 600.

74. In figure, AOC is a line, find x.

75. In figure, AOB is a line, determine x.

76. In figure, OA and OB are the opposite rays:

(i)If y = 1100, what is the value of x ?(ii)If x = 750, what is the value of y ?

77. In figure, AOC and BOC form a linear pair. If a - b= 800, find the values of a and b.

78. In figure, and intersect at O.

(i)Determine y when x = 600

(ii)Determine x when y = 400

79. In figure, lines AB, CD and EF intersect at O. Find themeasures of ÐAOC, ÐCOF.


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80. In figure, p, q and r are parallel lines intersected bytransversal t at X, Y and Z respectively. Find Ð1, Ð2 and

Ð3.81. In figure, OP and OQ are opposite rays. Find x.

82. In figure, ÐPOM and ÐQOM form a linear pair. If x - 2y =300, find x and y.

83. In figure, lines p and q intersect at O.If x = 350, find the values of y, z, w.

84. If fig ÐPQR = ÐPRQ, then prove that ÐPQS = ÐPRT

85. In fig, POQ is a line. Ray OR is perpendicular to line PQ.OS is another ray lying between rays OP and OR. Prove

that ÐROS = (ÐQOS - ÐPOS)

86. In fig. if AB || CD, EF CD and ÐGED - 1260, findÐAGE, ÐGEF and ÐFGE.

87. In fig AB || CD, ÐAPQ = 500 and ÐPRD = 1270, find xand y.

88. In figure, if x + y = w + z then prove that AOB is a straightline.


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89. In fig lines PQ and RS intersect each other at point O. IfÐPOR : ÐROQ = 5 : 7, find all the angles.

90. In fig AB || CD and CD || EF. Also EA AB. IfÐBEF =550, find the value of x, y and x:

91. In fig, if PQ||ST, ÐPQR = 1100 and ÐRST = 1300, findÐQRS.

92. In fig, Ðx = 620, ÐXYZ = 540. If YO and ZO are thebisectors of ÐXZY and ÐXZY respectively of DXYZ, findÐOZY and ÐYOZ.

93. In fig, if AB||DE, ÐBAC = 350 and ÐCDE = 530, findÐDCE.

94. In fig, if PQ PS, PQ||SR,ÐSQR = 280 and ÐQRT =650, then find the value of x and y.

95. Find the measure of the complementary angle of thefollowing angles :-(i) 220(ii) 63096. How many degrees are there in an angle which equalstwo-third of its complement?

97. Find the measure of the supplementary angle of thefollowing angles :(i) 450 (ii) 570

98. Two supplementary angles are in the ratio of 3 : 7. Findthe angles.

99. What value of x would make AOB a line in figure, ifÐAOC = 4x andÐBOC = (6x + 300)?

100. In fig, lines and intersect at O, formingangles as shown in the figure. If a = 350, find the valuesof b, c and d.

101. In fig, two straight lines PQ and RS intersect eachother at O. If ÐPOT = 750, find the value of a, b and c.


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102. In figure, lines AB and CD intersect at 0. If ÐAOC +ÐBOE = 700 and ÐBOD = 400, find ÐBOE and reflexÐCOE.

103. In figure, lines XY and MN intersect at O. If ÐPOY =900 and a : b = 2 : 3, find c.

104. In figure, OP bisects ÐBOC and OQ, ÐAOC. Provethat ÐPOQ = 900

105. Prove that the sum of all angles round a point isequal to 3600ORRays OA, OB, OC, OD and OE have the common initialpoint O. Show thatÐAOB + ÐBOC + ÐCOD + ÐDOE = ÐEOA = 3600

106. Prove that if a ray stands on a line, then the sum oftwo adjacent angles so formed is 1800

107. Prove that if the sum of two adjacent angles is 1800,then the non-common arms are two opposite rays.

108. In figure, find the values of x and y and then showthat AB || CD.

109. In figure, if AB || CD, CD || EF and y : z = 3 : 7, find x.(NCERT)

110. In figure, ÐABC = 650, ÐBCE = 300, ÐDCE = 350 andÐCEF = 1450. Prove that AB||EF.

111. In figure, if PQ||RS, ÐMXQ = 1350 and ÐMYR = 400,find ÐXMY.

112. In figure, if m||n and angle 1 and 2 are in the ratio 3: 2, determine all the angles from 1 to 8.


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113. Prove that if two parallel lines are intersected by atransversal, the bisector of any pair of alternate interiorangles is parallel.

114. If the bisectors of a pair of alternate interior angleare parallel, prove that given lines are parallel.

115. Prove that if two parallel lines are intersected by atransversal, then bisectors of any two correspondingangles are parallel.

116. If the bisectors of a pair of corresponding anglesformed by transversal are parallel, prove that given linesare parallel.

117. In a DABC, ÐB = 1050, Ðc = 500. Find ÐA.118. If the angles of a triangle are in the ratio 2 : 3 : 4,

determine three angles.119. In the fig, prove that p||m.

120. An exterior angle of a triangle is 1150 and one of theopposite angles is 350. Find the other two angles.

121. In figure, ÐACD = 1200 and ÐAPB = 1000, find x andy.

122. In figure, if OT PR, ÐTQR = 400 and ÐSPR = 300find x and y.

123. Two supplementary angles are in ratio 4 : 5, find theangles,124. If an angle differs from its complement by 10, findthe angle.125. In figure, OP and OQ bisects ÐBOC and ÐAOCrespectively. Prove that ÐPOQ = 900.

126. In figure, lines AB, CD and EF intersect at O. Findthe measures of ÐAOC, ÐDOE and ÐBOF

127. In figure if I║ m , n║ p and Ð1 = 850 find Ð2

128. The supplement of an angle is one third of itself.Determine the angle and its supplement.129. Two complementary angles are such that two timesthe measure of one is equal to three times measure

of the other. Find the measure of the large angle.130. Find the complement of each of the following angles.

(A) 36040’ (B) 42025’36”


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131. Write the supplementary angles of the following

anglels .

(A)(B) 980 35’ 20”

132. In figure, if ÐBOC = 7x + 200 and ÐCOA = 3x, thenfind the value of x for which AOB becomes a straight

line.

133. In figure, if x + y = w + z then prove that AOB is astraight line.

134. If the bisectors of two adjacent angles form a rightangle prove that their non common angles are in the

same straight line.135. In figure, find ÐCOD when ÐAOC + ÐBOD = 1000.

136. In figure x : y : z = 5 : 4 : 6. if XOY is a straight linefind the values of x, y and z.

137. In the given figure, AB is a mirror, PO is the incidentray and OR, the reflected ray. If ÐPOR = 1120 find

ÐPOA

138. In figure, if AB║ CD ║ EF and y : x = 3 : 7 find x

139. In figure if AB║ CD, EF CD and ÐGED = 1260,find ÐAGE, ÐGEF and ÐFGE.

140. ABC is an isosceles triangle in which ÐB = ÐV andL & M are points on AB & AC respectively such

that LM║ BC. If ÐA = 500 find ÐLMC.141. In figure if AB║ DF, AD║ FG, ÐBAC = 650, ÐACB =550. Find ÐFGH

142. In figure, AB║ ED and ÐABC = 300, ÐEDC = 700 thenfind x0.


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