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Chapter 6
Q1. Find the measure of an angle which is complement of itself.
Q2. Find the measure of an angle which forms a pair of supplementary angles with itself.
Q3. Two supplementary angles differ by 34°. Find the angles.
Q4. An angle is equal to five times its complement. Determine its measure.
Q5. An angle is equal to one-third of its supplement. Find its measure.
Q6. Two supplementary angles are in the ratio 2: 3. Find the angles.
Q7. Two supplement of an angle is one-third of itself. Determine the angle and its supplement.
Q8. In Fig. OA and OB are opposite rays:
(i)
lf x = 75, what is the value of y? (ii) lf y = 110, what is the value of x?
Q9. In Fig. , /AOC and /BOC form a linear pair. Determine the value of x.
Q10. In Fig. , /POR and /QOR form a linear pair. If a - b = 80, find the values of a and b.
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Q11. In Fig., OA, OB are opposite rays and /AOC + /BOD = 900 .Find /COD.
Q12. In Fig., OP bisects /BOC and OQ, /AOC . Show that /POQ = 900
Q13. In Fig. ray OE bisects /AOB and OF is a ray opposite to OE. Show that /FOB= /FOA
Q14. In Fig. lines XY and MN intersect at O. If /POY = 90° and a : b =2 : 3,find c.
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Q15. In Fig., /PQR= /PRQ, then prove that /PQS = /PRT.
Q16. If ray OC stands on line AB such that /AOE = /COB, then show that /AOC = 900
Q17. In Fig. , if
, then prove that AOB is a line.
Q18. It is given that / XYZ = 64° and XY is produced to a point P. Draw a figure from
the given information. If ray YQ bisects /ZYP, find /XYQ and reflex /QYP.
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Q19. In Fig. , lines 1 and
the values of b, c and d.
intesect at O, forming angles as shown in the figure. If a = 35°, find
Q20. In Fig. , determine the value of y.
Q21. In Fig. , three coplanar lines intersect in a common point, forming angles as shown.
Given a = 50° and b = 90°; find the values of c, d, e and f.
Q22. In Fig. , AB and CD are straight lines and OP and OQ are respectively the bisectors of
angles /BOD and / AOC. Show that the rays OP and OQ are in the same line.
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Q23. In Fig. , two straight lines PQ and RS intersect each other at O. If / POT = 750,find the
values of a, b and c.
Q24. In Fig. ,
/1= 65°. Find /5 and /8.
Q25. In Fig. find the values of x and y and then show that AB II CD.
Q26. In Fig., if AB II CD, CD II EF and y : z = 3 : 7, find x.
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Q27. In Fig. m II n and angles 1 and 2 are in the ratio 3: 2. Determine all the angles from 1 to 8.
Q28. In Fig. ,
are parallel lines intersected by a transversal p at X, Y and Z
respectively. Find / 1, / 2 and / 3. Give reasons.
Q29. In Fig. ,given that AB II CD.
(i) If /1 = (120 - x)O and /5 = 5xo, find the measures of /1 and /5.
(ii) If /4 = (x + 20)0 and /5 = (x + 8)°, find the measures of /4 and /5.
(iii) If /2 =(3x - 10)0 and /8 = (5x - 30)°, determine the measures of /2 and /8.
(iv) If /1 = (2x + y)O and /6 = (3x – y)o. determine the measures of /2 in terms of y.
(v) If /2 = (2x + 30)°, /4 = (x + 2y)o and /6 = (3y + 10)°, find the measure of /5.
(vi) If /2 = 2(/1), determine /7 .
(vii) If the ratio of the measures of /3 and /8 is 4 : 5, find the measures of /3 and /8,
(viii)If the complement of /5 equals the supplement of /4, find the measures of /4 and /5.
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Q30. In Fig., if AB II CD, EF
CD and /GED = 1260, find /AGE, / GEF and / FGE.
Q31. In Fig. , if AB II CD,/APQ = 50° and /PRD = 127°, find x and y.
Q32. In Fig , AB II DC and AD II BC. Prove that /DAB = /DCB.
Q33. In Fig. , AB CD. Determine /1
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Q34. In Fig. , AB CD. Determine x.
Q35. In Fig. , AB II CD. Find the value of x.
Q36. In Fig. , AB DE. Prove that /ABC +/BCD = 1800 + /CDE.
Q37. In Fig. , AB CD and EF II DQ. Determine / PDQ, / AED and / DEF.
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Q38. In Fig. , PQ RS, /PAB = 70° and /ACS = 100°. Determine /ABC, /BAC and /CAQ.
Q39. In Fig. , AB IICD and /F = 30°. Find / ECD.
Q40. In Fig. , OP II RS. Determine /PQR.
Q41. In Fig, if /2 = 120° and / 5 = 60°, show that m II n.
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Q42. In Fig. , show that AB II EF.
Q43. In Fig. , if /3 = 61° and /7 = 118°. Is m II n ?
Q44. In Fig. , given that /AOC = /ACO and /BOD = /BDO. Prove that AC II DB.
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Q45. In Fig., AB II CF and BC II ED. Prove that /ABC = / FDE.
Q46. In Fig., lines AB and CD are parallel and P is any point between the two lines. Prove that
/ABP+ /CDP= /DPB.
Q47. Prove that two lines perpendicular to the same line are parallel to each other.
Q48. If the bisectors of a pair of alternate angles formed by a transversal with two given
lines are parallel, prove that the given lines are parallel.
Q49. In Fig. AB II CD and CD II EF. Also, EA AB. If /BEF = 55°, find the values of x, y and z.
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Q50. In Fig.,if PQ II RS, /MXQ = 135° and /MYR = 40°, find /XMY.
Q51. In Fig. , PQ and RS are two mirrors placed parallel to each other. An incident ray
AB strikes the mirror PQ at B, the reflected ray moves along the path BC and strikes the mirror
RS at C and again reflects back along CD. Prove that AB II CD.
Theorem
1. To prove that If a ray stands on a line, then the Sum of the adjacent angles so formed is
180o
2. To prove that If the sum of two adjacent angles is 180°, then their non-common arms
are two opposite rays.
3. To Prove that The sum of all the angles round a point is equal to 360o.
4. To prove that If two lines intersect, then the vertically opposite angles are equal.
5. To prove that If a transversal intersects two parallel lines, then each pair of alternate
interior angles are equal.
6. To prove that If a transversal intersects two lines in such a way that a pair of alternate
interior angles are equal, then the two lines are parallel.
7. To prove that If a transversal intersects two parallel lines, then each pair of consecutive
interior angles are supplementary.
8. To prove that If a transversal intersects two lines in such a way that a pair of consecutive
interior angles are supplementary, then the two lines are parallel.
9. To prove that If two parallel lines are intersected by a transversal, the bisectors of any
pair of alternate interior angles are parallel.
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10. To prove that If two parallel lines are intersected by a transversal, then bisectors of any
two corresponding angles are parallel.
11. To prove that If the bisectors of a pair of corresponding angles formed by a transversal
with two given lines are parallel, prove that the given lines are parallel.
12. To prove that If a line is perpendicular to one of two given parallel lines, then it is also
perpendicular to the other line.
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