4 lines and angles - study pointstudy-point.com/spfile/14733316684.linesandanglesix.pdf · 57 lines...
TRANSCRIPT
54
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles
CHAPTER
4We are Starting from a Point but want to Make it a Circle of Infinite Radius
Lines and Angles
BASIC GEOMETRICAL CONCEPTS (AXIOMS, THEOREMS AND COROLLARIES)
Axioms The basic facts which are taken for granted, without proof, are called axioms.
Examples (i) Halves of equal are equal.
(ii) The whole is greater than each of its parts.
(iii) A line contains infinitely many points.
STATEMENTS A sentence which can be judged to be true or false is called a statement.
Examples (i) The sum of the angles of a triangle is 1800, is a true statement.
(ii) The sum of the angles of a quadrilateral is 1800, is a false statement.
(iii) x + 10 > 15 is a sentence but not a statement.
Theorems A statement that requires a proof, is called a theorem. Establishing the truth of a
theorem is known as proving the theorem.
Examples (i) The sum of all the angles around a point is 3600.
(ii) The sum of the angles of a triangle is 1800.
CAROLLARY : A statement, whose truth can easily be deduced from a theorem, is called its corollary.
EUCLID’S FIVE POSTULATES
1. A straight line may be drawn from any point to any other point.
2. A terminated line can be produced indefinitely.
3. A circle can be drawn with any center and any radius.
4. All right angle are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side of it taken
together less than two right angles, then the two straight lines, if produced indefinitely, meet on that
side on which the angles taken together are less than two right angles.
Later on the fifth postulate was modified as under.
‘For every line L and for every point P not lying on L, there exists a unique line M, passing through
P and parallel to L’.
Clearly, two distinct intersecting lines cannot be parallel to the same line.
SOME TERMS RELATED TO GEOMETRY
POINT A point is an exact location.
A fine dot represents a point.
We denote a point by a capital letter – A, B, P, Q, etc.
In the given figure, P is a point.
Line segment
The straight path between two points A and B is called the line segment AB .
The points A and B are called the end points of the line segment AB .
A line segment has a definite length.
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles55
The distance between two points A and B is equal to the length of the line segment AB .
RAY
A line segment AB when extended indefinitely in one direction is the ray
AB .
Ray
AB has one end point A.
A ray has no definite length.
A ray cannot be drawn, it can simply be represented on the plane of a paper.
To draw a ray would mean to represent it.
LINE A line segment AB when extended indefinitely in both the directions is called the line
AB .
A line has no end points. A line has no definite length. A line cannot be drawn, it can simply be represented
on the plane of a paper.
To draw a line would mean to represent it.
Sometimes, we lable lines by small letters l, m, n, etc.
INCIDENCE AXIOMS ON LINES
(i) A line contains infinitely many points.
(ii) Through a given point, infinitely many liens can be drawn.
(iii) One and only one line can be drawn to pass through two
given points A and B.
COLLINEAR POINTS
Three or more than three points are said to be collinear, if there is
a line which contains them all.
In the given figure A,B,C are collinear points, while P,Q,R are non-collinear.
INTERSECTING LINES
Two lines having a common point are called intersecting lines.
In the given figure, the lines AB and C intersect at a point O.
CONCURRENT LINES Three or more lines intersecting at the same point are said to be
concurrent.
In the given figure, lines l, m, n pass through the same point P and
therefore, they are concurrent.
PLANE
A plane is a surface such that every point of the line joining any
two points on it, lies on it.
Examples The surface of a smooth wall; the surface of the top of the table; the surface of a smooth
blackboard; the surface of a sheet of paper etc., are close examples of a plane. These
surfaces are limited in extent but the geometrical plane extends endlessly in all
directions.
Parallel Lines Two lines l and m in a plane are said to be parallel, if they have no point in common and we write, l || m.
The distance between two parallel lines always remains the same.
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles56
Questions
1. (i) How many lines can be drawn to pass through a given point?
(ii) How many lines can be drawn to pass through two given points?
(iii) In how many points can the two lines at the most intersect?
(iv) If A, B, C are three collinear points, name all the line segments determined by them.
2. Which of the following statements are true?
(i) A line segment has no definite length.
(ii) A ray has no end point.
(iii) A line has a definite length.
(iv) A line
AB is the same as line
BA .
(v) A ray
AB is the same as ray
BA .
(vi) Two distinct points always determine a unique line.
(vii) Three lines are concurrent if they have a common point.
(viii) Two distinct lines cannot have more than one point in common.
(ix) Two intersecting liens cannot be both parallel to the same line.
(x) Open half-line OA is the same thing as ray
OA .
(xi) Two lines may intersect in two points.
(xii) Two lines l and m are parallel only when they have no point in common.
ANGLES AND THEIR PROPERTIES
ANGLE
Two rays OA and OB having a common end point O form angle
AOB, written as AOB .
OA and OB are called the arms of the angle and O is called its
vertex.
INTERIOR OF AN ANGLE
The interior of AOB is the set of all points in its plane, which lie
on the same side of OA as B and also on the same side of OB as
A, e.g., P is a point in the interior of AOB . Any point on any
arm or vertex is said to lie on the angle, e.g., Q is a point on
AOB .
EXTERIOR OF AN ANGLE
The exterior of an angle AOB is the set of all those points in its
plane, which do not lie on the angle or in its interior. In the given
figure, R is a point in the exterior of AOB .
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles57
MEASURE OF AN ANGLE
The amount of turning from OA to OB is called the measure of
AOB , written as m AOB . An angle is measured in degrees
denoted by.
AN ANGLE OF 3600
If a ray OA starting from its original OA, rotates about O, in the
anticlockwise direction and after making a complete revolution it
comes back to its original position, we say that it has rotated
through 360 degrees, written as 3600.
This complete rotation is divided into 360 equal parts. Each part measures 10.
10 = 60 minutes, written as 60’.
1’ = 60 seconds, written as 60’’.
We use a protractor to measure an angle.
KINDS OF ANGLE
(i) RIGHT ANGLE An angle whose measure is 900 is called a right angle.
(ii) ACUTE ANGLE An angle whose measure is more than 00 but less than 90
0 is called an acute angle.
(iii) OBTUSE ANGLE An angle whose measure is more than 900 but less than 180
0 is called an obtuse
angle.
(iv) STRAIGHT ANGLE An angle whose measure is 1800 is called a straight angle.
(v) REFLEX ANGLE An angle whose measure is more than 1800 but less than 360
0 is called a reflex
angle.
(vi) COMPLETE ANGLE An angle whose measure is 3600 is called a complete angle.
EQUAL ANGLES
Two angles are said to be equal, if they have the same measure.
Bisector of an angle A ray OC is called the bisector of AOB , if
m AOC = m BOC .
In this case, AOC = BOC = 2
1AOB
COMPLEMENTARY ANGLES
Two angles are said to be complementary, if the sum of their measures is 900.
Two complementary angles are called the complement of each other.
Example Angles measuring 550 and 35
0 are complementary angles.
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles58
SUPPLEMENTARY ANGLES
Two angles are said to be supplementary, if the sum of their measures is 1800.
Example Angles measuring 620 and 118
0 are supplementary angles.
Example 1 Find the measure of an angle which is 240 more than its complement.
Solution Let the measure of the required angle be x0.
Then measure of its complement = (90 – x)0.
x – (90 – x) = 24 2x = 114 x = 57.
Hence, the measure of the required angle is 570.
Example 2 Find the measure of an angle which is 320 less than its supplement.
Solution Let the measure of the required angle be x0.
Then measure of its complement = (180 – x)0.
x – (180 – x) = 32 2x = 148 x = 74.
Hence, the measure of the required angle is 740.
Example 3 Find the measure of an angle, if six times its complement is 120 less than twice its
supplement.
Solution Let the measure of the required angle be x0.
Then measure of its complement = (90 – x)0.
Measure of its supplement = (180 – x)0
6 (90 – x) = 2 (180 – x) – 12 540 – 6x = 360 – 2x – 12
4x = 192 x = 48.
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles59
IX ACADEMIC QUESTIONS Subjective
Assignment – 1
1. Define the following terms:
(i) Angle (ii) Interior of an angle
(iii) Obtuse angle (iv) Reflex angle
(v) Complementary angles (vi) Supplementary angles
2. Find the complement of each of the following angles
(i) 580 (ii) 16
0 (iii)
2
1 of a right angle
3. Find the supplement of each of the following angles.
(i) 630 (ii) 138
0 (iii)
5
3 of a right angle
4. Find the measure of an angle which is 360 more than its complement.
5. Find the measure of an angle which is 250 less than its supplement.
6. Two supplementary angles are in the ratio 3 : 2. Find the angles.
7. Find the measure of an angle, if seven times its complement is 100 less than three times its supplement.
8. In Fig. lines PQ and RS intersect each other at point O. If POR : ROQ = 5 : 7, find all the angles.
9. In Fig. ray OS stands on a line POQ. Ray OR and ray OT are angle bisectors of POS and SOQ,
respectively. If POS = x, find ROT.
P Q
R S
O
T
10. In Fig. OP, OQ, OR and OS are four rays. Prove that POQ + QOR +
SOR +POS = 360°.
P
R
S
Q
O
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles60
11. 11. In Fig. lines AB and CD intersect at O. If AOC + BOE = 70° and BOD = 40°, find BOE
and reflex COE.
12. 12. In Fig. lines XY and MN intersect at O. If POY = 90° and
a : b = 2 : 3, find c.
13. In Fig. PQR = PRQ, then prove that PQS = PRT.
14. In Fig. if x + y = w + z, then prove that AOB is a line.
15. 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 ROS =2
1(QOS – POS).
16. It is given that XYZ = 64° and XY is produced to point
P. Draw a figure from the given information. If ray YQ
bisects ZYP, find XYQ and reflex QYP.
SOME ANGLE RELATIONS
ADJACENT ANGLES Two angles are called adjacent angles, if
(i) they have the same vertex,
(ii) they have a common arm and
(iii) their non-common arms are on either side of the common arm.
In the given figure, AOC and BOC are adjacent angles
having the same vertex O, a common arm OC and their non-
common arms OA and OB on either side of OC.
LINEAR PAIR OF ANGLES
Two adjacent angles are said to form a linear pair of angles, if
their non-common arms are two opposite rays.
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles61
In the adjoining figure, AOC and BOC are two adjacent
angles whose non-common arms OA and Ob are two opposite
rays, i.e., BOA is a line.
AOC and BOC form a linear pair of angles.
SOME RESULTS ON ANGLES RELATIONS
Theorem 1 If a ray stands on a line then the sum of the adjacent angles so formed is 1800.
Given A ray CD stands on a line AB such that ACD and BCD are formed.
To prove 0180 BCDACD
Construction Draw CE AB.
Proof ECDACEACD (i)
and ECDBCEBCD (ii)
Adding (i) and (ii), we get :
BCDACD = )()( ECDBCEECDACE
= BCEACE
= (900 + 90
0) = 180
0 [ 090 BCEACE ]
Hence, 0180 BCDACD
REMARK we may state the above theorem as the sum of the angles of a linear pair is 1800.
COROLLARY 1 Prove that the sum of all the angles formed on the same side of a line at a given point on
the line is 1800.
Given AOB is a straight line and rays OC, OD and OE stand on it, forming DOECODAOC ,, and
EOB .
To prove 0180 EOBDOECODAOC .
Proof Ray OC stands on line AB.
COBAOC = 1800.
0180)(( EOBDOECODAOC
[ EOBDOECODCOB ]
0180 EOBDOECODAOC
Hence, the sum of all the angles formed on the same side of
line AB at a point O on it is 1800.
COROLLARY 2
Prove that the sum of all the angles around a point is 3600.
Given A point O and the rays OA, OB, OC, OD and OE make angles
around O.
To Prove 0360 EOADOECODBOCAOB .
Construction Draw a ray OF opposite to ray OA.
PROOF Since ray OB stands on line FA, we have :
BOFAOB = 1800. [linear pair]
COFBOCAOB = 1800. (i)
[ COFBOCBOF ]
Again, ray OD stands on line FA.
DOAFOD = 1800. [linear pair]
or EOADOEFOD = 1800.
[ EOADOEDOA ]
Adding (i) and (ii), we get :
EOADOEFODCOFBOCAOB = 3600.
0360 EOADOECODBOCAOB
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles62
[ CODFODCOF ]
Hence, the sum of all the angles, around a point O is 3600.
VERTICALLY OPPOSITE ANGLES
Two angles are called a pair of vertically opposite angles, if their
arms form two pairs of opposite rays.
Let two lines AB and CD intersect at a point O. Then, two pairs of
vertically opposite angle are formed :
(i) BODandAOC (ii) BOCandAOD
Theorem 2 If two lines intersect then the vertically opposite angle are equal.
Given Two lines AB and CD intersect at a point O.
To Prove (i) BODAOC , (ii) BOCAOD
PROOF Since ray OA stands on line CD, we have
0180 BODAOC [linear pair]
Again ray OD stands on line AB.
0180 BODAOD [linear pair]
BODAODAODAOC [each equal to 1800]
BODAOC
Similarly, BOCAOD
Example 4 In the adjoining figure, AOB is a straight line. Find AOC and BOD .
Solution Since AOB is a straight line, the sum of all the angles on the same side of AOB at a point O
on it, is 1800.
x + 65 + (2x – 20) = 180
3x = 135 x = 45.
045AOC and BOD = (2 45 – 20)
0 = 70
0.
Example 5 In the adjoining figure, what value of x will make AOB a straight line?
Solution AOB will be a straight line, if 0180 BOCOCA .
(3x + 5) + (2x – 25) = 180
5x = 200 x = 40.
Hence, x = 40 will make AOB a straight line
THE ANGLES FORMED WHEN A TRANSVERSAL CUTS TWO LINES
Let AB and CD be two lines, cut by a transversal t. Then, the
following angles are formed
(i) Pairs of corresponding angles : ( 5,1 ); ( 8,4 ); (
6,2 ) and ( 7,3 ).
(ii) Pairs of alternate interior angles : ( 5,3 ) and ( 6,4 )
(iii) Pairs of consecutive interior angles (allied angles or
conjoined angles) : ( 5,4 ) and ( 6,3 ).
REMARKS We shall abbreviate as follows :
(i) Corresponding Angles as corres s
(ii) Alternate Interior Angles as Alt. Int. s
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles63
(iii) Consecutive Interior Angles as Co. Int. s
CORRESPONDING ANGLES AXIOM
If a transversal cuts two parallel lines then each pair of corresponding angles are equal.
Conversely, if a transversal cuts two lines, making a pair of corresponding angles equal, then the lines are
parallel.
Thus, whenever AB || CD are cut by a transversal t, then 7362;84;51 and .
On the other hand, if a transversal t cuts two lines AB and CD such that ( 51 ) or ( 84 ) or (
62 ) or ( 73 ) then AB || CD.
Theorem 3 If a transversal intersects two parallel lines then alternate angles of each pair of interior
angles are equal.
Given AB || CD and a transversal t cuts AB at E and CD at F, forming two pairs of alternate interior angles,
namely ( 5,3 ) and ( 6,4 )
To Prove 53 and 64 .
PROOF We have 13 (vert. opp. s ) and 51 (corres s )
53 .
Again, 24 (vert. opp. s )
and 62 (corres s )
64 .
Hence, 53 and 64 .
Theorem 4 If a transversal intersects two parallel lines then
each pair of consecutive interior angles are
supplementary.
Given AB || CD and a transversal t cuts AB at E and CD at F,
forming two pairs of consecutive interior angles, namely (
63 ) and ( 54 ).
TO PROVE 63 = 1800 and
018054 .
PROOF Since ray EF stands on line AB, we have 018043 (linear pair).
But, 64 (Alt. Int. s )
018063
Again, since ray FE stands on line CD,
We have 018056 .
But, 46 (Alt. Int. s )
018054 .
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles64
Hence, 018063 and
018054 .
Theorem 5 (Converse of Theorem 1) if a transversal intersects two lines, making a pair of alternate interior angles
equal, then the two lines are parallel.
Given A transversal t cuts two lines AB and CD at E and F
respectively such
To Prove AB || CD.
PROOF We have 53 (given)
But, 13 (vert. opp. s )
51 .
But, these are corresponding angles.
AB || CD (by corres. s axiom)
Theorem 6 (Converse of Theorem 2) If a transversal intersect two lines in such a way that a pair of
consecutive interior angles are supplementary then the two lines are parallel.
Given A transversal cuts two lines AB and CD at E and F respectively such that 018054
To Prove AB || CD.
PROOF Since ray EB stands on line t, we have
018041 (linear pair)
and 018054 (given)
5441
This gives, 51 .
But, these are corresponding angles.
AB || CD (by corres. s axiom)
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles65
IX ACADEMIC QUESTIONS Subjective
Assignment – 2
1. Prove that the bisectors of the angles of a linear pair are at right angles.
2. In the adjoining figure, AOB is a straight line. Find the value of x.
3. In the adjoining figure, AOB is a straight line. Find the
value of x. Hence, find CODOCA , and BOD .
4. In the adjoining figure, x : y : z = 5 : 4 : 6. If XOY is a
straight line, find the values of x, y and z.
5. 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.
6. In the given figure, AB || CD. Find the value of x.
7. In the given figure, AB || CD. Find the value of x.
8. For what value of x will the lines l and m be parallel to each
other?
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles66
9. In Fig. if PQ || RS, MXQ = 135° and MYR = 40°, find
XMY.
10. If a transversal intersects two lines such that the bisectors of a pair
of corresponding angles are parallel, then prove that the two lines
are parallel.
11. In Fig. AB || CD and CD || EF. Also EA AB. If BEF =
55°, find the values of x, y and z.
12. In Fig. if AB || CD, CD || EF and y : z = 3 : 7, find x.
13. In Fig. if AB || CD, EF CD and GED = 126°, find
AGE, GEF and FGE.
14. In Fig. if PQ || ST, PQR = 110° and RST = 130°, find
QRS.
[Hint : Draw a line parallel to ST through point R.]
15. In Fig. if AB || CD, APQ = 50° and PRD = 127°, find
x and y.
16. In Fig. if QT PR, TQR = 40° and SPR = 30°, find x and y.
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles67
17. In Fig. sides QP and RQ of PQR are produced to points S and T respectively. If SPR = 135°
and PQT = 110°, find PRQ.
18. In Fig. X = 62°, XYZ = 54°. If YO and ZO are the bisectors of XYZ and XZY
respectively of XYZ, find OZY and YOZ.
19. In Fig. if AB || DE, BAC = 35° and CDE = 53°, find DCE.
20. In Fig. 6.42, if lines PQ and RS intersect at point T, such that PRT = 40°, RPT = 95° and
TSQ = 75°, find SQT.
21. In Fig. 6.43, if PQ PS, PQ || SR, SQR = 28° and
QRT = 65°, then find the values of x and y.
22. In Fig. 6.44, the side QR of PQR is produced to a point
S. If the bisectors of PQR and PRS meet at point T,
then prove that QTR = 2
1QPR.
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles68
XI SCIENCE & DIP. ENTRANCE Multiple Choice Question
Assignment – 3
1. In the figure the lines AB and BD lie in a straight line. If 8x3ABC
and 4xDBC , then what is the value of x ?
(a) 044 (b)
052 (c) 060 (d)
064
2. In the above figure if 3 ABC2DBC , then what is value of DBC
?
(a) 036 (b)
048 (c) 072 (d)
081
3. In the figure if POR and QOR form a linear pair. If 080ba ,
then angle ''a is
(a) 080 (b)
0130 (c) 0140 (d)
0150
4. In the figure AB is a straight line. If ,20xAOC0
015x2COD and 010xBOD , then COD is
(a) 045 (b)
0180 (c) 075 (d)
090
5. In the above figure if 2:4:3BOD:COD:AOC , then AOC is
(a) 040 (b)
050 (c) 060 (d) 70
6. In the following figure 090AOB and CD is a straight line. If
012xy , then z is equal to
(a) 075 (b)
090 (c)0100 (d)
0102
7. In the above Fig if 0105z what is the value of
0y ?
(a) 020 (b)
040 (c) 060 (d)
075
8. In the figure 090BOE AB and CD are straight lines. If
3:2: yx , then the value of ‘ z ’ is
(a) 076 (b)
060 (c) 0118 (d)
0126
9. In the above figure if xz 3 , then ‘ y ’ is equal to
(a) 045 (b)
060 (c) 075 (d)
090
10. In the figure AB and CD are straight lines. If mx and 090n , then
the value of 0z is
(a) 045 (b)
0120 (c) 0135 (d)
0145
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles69
11. In the above figure if 040y and 10m2n , then the value of
0x
is
(a) 040 (b)
050 (c) 060 (d)
070
12. In the figure, the value of ‘ x ’ is
(a) 30o (b) 40
o (c) 45
o (d)
050
13. In the figure, AOB is a straight line OP and OQ are bisectors of
BOC and AOC respectively, then the value of POQ is
(a) 070 (b)
032 (c) 90o (d)
0120
14. What is the value of ‘ x ’ in the figure?
(a) 60o
(b) 45o
(c) 30o
(d) 15o
15. What is the complement angle of 082 ?
(a) 79o (b)
050 (c) 030 (d)
08
16. What is the supplement angle of 0123 ?
(a) 1230o (b)
060 (c) 057 (d)
045
17. If angles 010a2 and 011a are complementary angles then ‘ a ’
is equal to
(a) 025 (b)
037 (c) 045 (d)
055
18. An angle is 014 more than its complement. What is its measure?
(a) 52o (b)
055 (c) 062 (d)
075
19. The measure of an angle is twice the measure of its supplementary
angle. What is the value of greatest angle?
(a) 060 (b)
090 (c) 0120 (d)
0130
20. When two supplementary angles differ by 032 , what is the value of
smaller angle?
(a) 040 (b)
045 (c) 060 (d)
074
21. How many degrees are there in an angle which equals one-fifth of its
complement?
(a) 18o (b)
015 (c) 012 (d)
010
22. How many degrees are there in an angle which equals two-third of its
supplement?
(a) 072 (b)
090 (c) 0120 (d)
0130
23. How many degrees are there in an angle whose complement is one-
fourth of its supplement?
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles70
(a) 030 (b)
060 (c) 045 (d)
090
In the figure PQ is an incident ray and QR the reflected ray. If 0124PQR , then RQB is
(a) 025 (b)
028 (c) 032 (d)
036
24. The measure of an angle whose supplement is three-times as large as
its complement is
(a) 030 (b)
045 (c) 060 (d)
075
25. If two angles are complementary of each other, then each angle is
(a) An obtuse angle (b) A right angle
(c) An acute angle (d) A supplementary angle
26. In the figure if AOB is a straight line, then the value of ‘ x ’ is
(a) 90o (b)
045 (c)
0
2
122 (d)
0150
27. An angle is greater than 0180 but less than
0360 is called
(a) An acute angle (b) An obtuse angle
(c) An adjacent angle (d) A reflex angle
28. If CD||AB and 10b5a2 then b is equal to
(a) 050 (b)
060 (c) 075 (d)
080
29. In the given figure, where AB||CD||EF if 015z2x , then what is
the value of y?
(a) 060 (b)
090 (c) 0120 (d)
0130
30. From the figure, where AB||CD, what is the value of x?
(a) 0180 (b)
0210 (c) 0240 (d)
0260
31. From the figure calculate the value of x
(a) 30o
(b) 45o
(c) 60o
(d) 75o
32. If AB||CD as shown in the figure, calculate the value of x
(a) 050 (b)
060
(c) 0250 (d)
0320
33. In the adjoining figure 0100ABC ,
0120EDC and AB||DE, then
BCD is equal to
(a) 040 (b)
060
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles71
(c) 080 (d)
0100
34. In the given figure If AB||CD, then FXE is equal to
(a) 030 (b)
050
(c) 070 (d)
080
35. If two parallel lines are intersected by a transversal line then the
bisectors of the interior angles form a
(a) Rhombus (b) Parallelogram
(c) Square (d) Rectangle
36. In the figure AB||CD 0100BAE and
025AEC , what is the
value of DCE ?
(a) 0100 (b)
0125
(c) 075 (d)
0160
37. In the figure m||n and p||q. If 0751 , then 2 is equal to
(a) 0105 (b)
090
(c) 075 (d)
060
38. Two parallel lines AB and CD are intersecting by a transversal line EF
at M and N respectively. The lines MP and NP are the bisectors of
interiors angles BMN and DMN on the same side of transversal
line, then MPN is equal to
(a) 045 (b)
060 (c) 090 (d)
0120
39. In the figure arms BA and BC of ABC are respectively parallel to
arms ED and EF of DEF , then DEFABC is
(a) 090 (b)
0120
(c) 0160 (d)
0180
011-26925013/14+91-9811134008+91-9582231489
NTSE, NSO Diploma, XI Entrance
Lines and Angles72
ANSWER
Assignment – 1
2. (i) 320 (ii) 74
0 (iii) 45
0
3. (i) 1170 (ii) 42
0 (iii) 126
0
4. 630 5. 77.5
o 6. 108
o, 72
o 7. 25
o
8. POR = QOS = 750, ROQ = POS = 105
0 9. 90
o
11. BOE = 30o and reflex COE = 250
o 12. c = 126
o
16. QYP = 302o
Assignment – 2
2. 118 o 3. x = 32
o, AOC = 103
o, COD = 45
o, BOD = 32
o
4. x = 60o, y = 48
o, z = 72
o 6. x = 40
o 7. x = 50
o 8. (i) x = 30
o (ii) x = 5
o
9. XMY = 85o 11. x = 130
o, y = 130
o, z = 4
o 12. x = 126
o
13. AGE = 126o, GEF = 36
o, FGE = 54
o 14. QRS = 60
o 15. x = 50
o, y = 77
o
16. x = 50o, y = 80
o 17. 65
o 18. 32
o, 121
o 19. 92
o
20. 60o 21. 37
o, 53
o
Assignment – 3
1.a 2.c 3.b 4b 5.c 6.d 7.b 8.d 9.a 10.c 11.b 12.b
13.c 14.d 15.d 16.c 17.b 18.a 19.c 20.d 21.b 22.a 23.b 24.b
25.b 26.c 27.b 28.d 29.a 30.d 31.d 32.a 33.c 34.a 35.d 36.d
37.b 38.a 39.c 40.d