A part of a line with two end points is called line segment and is denoted as AB

HTR 6 Lines and ngles Introduction In previous class we have studied that minimum two points are required to draw a line. line having one end point is called a ray. Now if two rays originate from a point, an angle is formed. If two lines intersect each other, different angles are formed. Here we will study different properties related to lines and angles. asic Terms and efinitions Line Segment part of a line with two end points is called line segment and is denoted as Ray part of a line with one end point is called ray and is denoted as point. It can be etended further from Line It can be etended from both sides (left and right) and is denoted as ollinear oints Three or more points are said to be collinear if a single straight line passes through them. Here,, are collinear. Non-ollinear oints Three or more points not lying on a single straight line are called non-collinear points.,, are not collinear. Mathematics lass 9 LINS N NGLS 99

ngle When two rays originates from the same end point they form an angle. Rays (arms of angle) nd point (Verte) ngle The rays are called arms of an angle and end point is called verte. Types of ngles Intersecting Lines and Non-Intersecting Lines In figure (i) and are intersecting lines. In figure (ii) and are non-intersecting lines (parallel lines). In parallel lines the lengths of the common perpendiculars at different points are equal. This equal length is called the distance between two parallel lines. airs of ngles z omplementary ngles If the sum of measure of two angles is 90, they are known as complementary angles. or eample : Q + = 70 + 0 = 90 \ Q and are complementary angles and Q is called complement of and vice-versa. 70 Q 0 z Supplementary ngles If the sum of measure of two angles is 80, they are called supplementary angles. or eample : XY and QR are supplementary as XY + QR = 70 + 0 = 80 and XY is called supplement of QR and vice-versa. z djacent ngles X 70 Y 0 Q R Two angles are said to be adjacent angles or adjacent to each other if (i) They have common arm. 00 LINS N NGLS lass 9 Mathematics

(ii) They have common verte. (iii) Non-common arms lying on the different sides of the common arm. In the figure, and are adjacent angles. They have common verte, common arm and non-common arms and lying on the different sides of. z Linear air Two adjacent angles whose sum is 80 are said to form linear pair or in other words, supplementary adjacent angles are called linear pair. Here, + = 80, so they form linear pair. Linear air iom iom- If a ray stands on a line, then the sum of two adjacent angles so formed is 80. Ray stands on line, then and are adjacent angles. + = 80 iom- If the sum of two adjacent angles is 80, then the non-common arms of the angles form a straight line. Suppose and are two adjacent angles, with common arm, non common arms and. If + = 80, then would be a straight line. Vertically pposite ngles If two lines and intersects each other at point, then four angles are formed. is vertically opposite to. Similarly is vertically opposite to. These are called pairs of vertically opposite angles. Theorem Statement : If two lines intersect each other, then the vertically opposite angles are equal. Given : and are two lines intersecting each other at point. is vertically opposite to and is vertically opposite to. To rove : = = roof : + = 80...() [Linear pair] + = 80...() [Linear pair] rom () and (), we get + = + = Similarly, = Mathematics lass 9 LINS N NGLS 0

is a line such that = 5 and = 4. ind the value of and hence find and. 5 Soln.: Since is a line, \ and form a linear pair + = 80 5 + 4 = 80 9 = 80 = 0 4 \ = 5 = 5 0 = 00 = 4 = 4 0 = 80 If the supplement of an angle is two-third of itself. etermine the angle and its supplement. Soln.: Let the angle be \ Its supplement = of = 3 3 3+ + = 80 = 80 3 3 5 80 3 = 80 = = 08 3 5 \ The angle is 08 and its supplement = 08 = 36 = 7 3 3 Lines, and intersect at. ind the measures of, and 40 35 Soln.: and intersect at point. \ = [Vertically opposite angles] \ = 40 Similarly, = [Vertically opposite angles] = 35 lso, + + = 80 [Straight angle] = 80 75 = 05 4 Lines l and l intersects at point forming angles a, b, c, d. l a d c If a = 45, find b, c, d, a + b, b + c and verify that a + d = b + c. Soln.: a = 45 c = 45 [Vertically opposite angles] a + d = 80 [Linear pair] d = 80 a = 80 45 = 35 d = b [Vertically opposite angles] b = 35 Now, a + d = 45 + 35 = 80 b + c = 35 + 45 = 80 a + d = b + c b l Transversal line which intersects two or more lines at distinct points is called a transversal. l l q m q m ( q is a transversal of l and m) ( q is not a transversal of l and m) 0 LINS N NGLS lass 9 Mathematics

ngles ormed by a Transversal Suppose l and m are two parallel lines intersected by a transversal n. Interior ngles, 4, 5, 6 terior ngles 3,, 7, 8 orresponding ngles : and 8, and 6, 3 and 5, 4 and 7 [Note that pairs of angles lie either above or below the lines.] lternate Interior ngles : 4 and 6, and 5 lternate terior ngles : 3 and 8, and 7 n 3 4 5 6 7 8 o-interior ngles : (lso called consecutive interior angles or allied angles) 4 and 5, and 6. o-terior ngles : and 8, 3 and 7. l m Results when a Transveral Intersects Two arallel lines orresponding ngles iom (i) If a transversal intersects two parallel lines, then each pair of corresponding angles are equal. (ii) If a transversal intersects two lines such that a pair of corresponding angles are equal, then the two lines are parallel to each other. lternate Interior ngles Theorem (i) If a transversal intersects two parallel lines, then each pair of alternate interior angles are equal. (ii) If a transversal intersects two lines such that a pair of alternate interior angles are equal, then the two lines are parallel. o-interior ngles Theorem (i) If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary. In other words, co-interior angles are supplementary. (ii) If a transversal intersects two lines such that a pair of co-interior angles are supplementary, then the two lines are parallel. o-eterior ngles Theorem (i) If a transversal intersects two parallel lines then each pair of co-eterior angles are supplementary. (ii) If a transversal intersects two lines such that a pair of co-eterior angles are supplementary, then two lines are parallel. arallel Lines Theorem Lines which are parallel to the same line are parallel to each other. This means if l n and m n l m. l m n Mathematics lass 9 LINS N NGLS 03

5 In the given figure if l m and t is a transversal, determine. ( + 6 ) l Q = 68 [rom (ii)] So, Q = = 68 and = 34 Soln.: l m t 00 + 6 + 00 = 80 [o-eterior angles] = 80 6 = 64 = 3 6 and Q. etermine Q, and. 34 Soln.: and transveral intersects them at and respectively. \ = [orresponding angles] Q 78 = 34... (i) Now ray stands on at \ + = 80 [Linear pair] + + = 80 m [ = + ] 34 + + 78 = 80 [rom (i)] = 80 = 68...(ii) Now Q and transversal intersects them at and respectively. \ = Q [orresponding angles] 7 If l m n and Q RS, find QRS. 70 Q 5 4 3 R Soln.: tend Q to point and SR to point. l m = 5 [lternate interior angles] and = 70 lso, 3 = 3 = 70 S l m n [orresponding angles] [ S, corresponding angles] Now, 3 + 4 = 80 [Linear pair] 70 + 4 = 80 4 = 0 \ QRS = + 4 = 5 + 0 = 35 8 In figure, if, G = 65 and = 80, then find and y. 65 80 y G Soln.: = [lternate interior angles] 65 + = 80 = 80 65 = 5 Now, G = G [lternate interior angles] y = 65. roperties of Triangles z ngle Sum roperty of a Triangle Theorem Statement : The sum of the three angles of a triangle is 80. i.e., + + 3 = 80 04 LINS N NGLS lass 9 Mathematics

roof : raw a line XY through point parallel to. 3 = 4 [lternate interior angles] X 5 = [lternate interior angles] lso 5 + + 4 = 80 [Straight angle] Replacing 5 and 4 by and 3 respectively, we get + + 3 = 80 + + 3 = 80 \ Sum of all angles of a triangle is 80. 5 4 3 Y z terior ngle roperty of a Triangle Theorem 3 Statement : If a side of a triangle is produced, then the eterior angle so formed is equal to the sum of the two interior opposite angles. Given : In, side is etended to point. is an eterior angle. To rove : = + roof : + + = 80...() [ngle sum property of triangle] lso, + = 80...() [Linear pair] rom () and (), we get + + = + = + 9 If one angle of a triangle is 7 and the difference of the other two angles is, find the other two angles. Soln.: ne angle of the triangle = 7 Let other two angles be and + [Q The difference between the two angles is ] So, + + + 7 = 80 [ngle sum property of triangle] + 84 = 80 = 80 84 = 96 96 = = 48 \ The other angles are 48 and 48 + = 60 0 In the given figure show that l m l m 73 4 3 n Soln.: + + = 80 [ngle sum property of triangle] + 3 + 4 = 80 = 80 73 = 07 Mathematics lass 9 LINS N NGLS 05

+ = 80 [Linear pair] = 80 07 = 73 Since, = 73 and = 73 So, = l m [s corresponding angles are equal] The sides, and of, are produced in order, forming eterior angles, and. rove that + + = 360. Soln.: 3 rom eterior angle property of triangle, = + 3...(i) = + 3...(ii) = +...(iii) dding (i), (ii) and (iii), we get + + = + 3 + + 3 + + = ( + + 3) = 80 [y angle sum property of triangle] + + = 360 06 LINS N NGLS lass 9 Mathematics

ngle n angle is formed when two rays originate from the same point. Some ngle Relations z djacent ngles Two angles are called adjacent angles, if (i) they have same verte (ii) they have common arm (iii) uncommon arms are on different sides of the common arm z Linear air of nlges Two adjacent angles are said to form a linear pair of angles, if (i) their non-common arms are two opposite rays (ii) the sum of the adjacent angles so formed is 80 (iii) they are supplementary. z Vertically pposite ngles If two lines and intersects each other at point, then is vertically opposite to. Similarly is vertically opposite to. z Transversal line which intersects two or more given lines at distinct points, is called a transversal of the given lines. z The measures of vertical opposite angles are equal. z The measures of angles forming a linear pair has total 80. z ngle Relationships ormed by arallel Lines being cut by a Transversal. (i) The measures of alternate eterior angles are equal. (ii) The measures of alternate interior angles are equal. (iii) The measures of the same-side interior angles are supplementary (sum to 80 ) (iv) The measures of corresponding angles are equal z Lines which are parallel to a given line are parallel to each other. z Sum of three angles of a triangle is 80. z If a side of a triangle is produced, the eterior angle so formed is equal to the sum of the two interior opposite angles. Mathematics lass 9 LINS N NGLS 07

SLV XMLS. rove that the sum of all angles formed on the same side of a line at a given point on the line is 80. Soln.: Given : is a straight line and rays, and stands on it, forming,, and. To rove : + + + = 80. roof : Ray stands on line. \ + = 80 [Linear pair] + ( + + ) = 80 + + + = 80. Hence, the sum of all the angles formed on the same side of line at a point on it is 80.. rove that the bisectors of the angles of a linear pair are at right angle. Soln.: Given : and form a linear pair of angles. and are the bisectors of and respectively. To rove : = 90. roof : + = 80 [Linear pair] + = 90 + = 90 [Q and are the bisectors of and ] = 90. Hence, the bisectors of the angles of a linear pair are at right angle. 3. rove that the bisectors of a pair of vertically opposite angles are in the same straight line. Soln.: Given : Two lines and intersecting each other at a point. lso, and are the bisectors of and respectively. To rove : is a straight line. roof : Since, the sum of all angles around a point is 360, \ + + + = 360 + + = 360 [Q = (Vertically opposite angles). is bisector of, is bisector ] + + = 80 =80 Hence, is a straight line. 4. In the figure, lines and intersect at. If + = 80 and = 30, find and refle. Soln.: + = 80...(i) [Given] = 30...(ii) [Given] Lines and intersect at. \ = [Vertically opposite angles] = 30 [rom (ii)] Now, putting the value of in (i), we have 30 + = 80 = 80 30 = 50. lso, + = 80 [Linear pair] = 80 30 = 50 and refle = 360 50 = 0. 08 LINS N NGLS lass 9 Mathematics

5. In the figure, lines XY and MN intersect at. If Y = 90 and a = b, find c. 7. If, then find the value of. M X b a c Y Soln.: onstruct a line l through parallel to. N Soln.: Given a = b Let a = then b = XM + M + Y = 80 [Linear pair] + + 90 = 80 + 90 = 80 = 90 90 = = 45 \ XM = b = 45 and M = a = 45 Now, XN = c = MY [Vertically opposite angles] = M + Y = 45 + 90 Hence, c = 35. 6. In the figure, if, and y : z = : 3, find. G Soln.: Let y = a and z = 3a HI + IH = 80 [, o-interior angles] y + z = 80 a + 3a = 80 5a = 80 80 a = = 36 5 \ y = a = 36 = 7 and z = 3a = 3 36 = 08 lso, and GI is a transversal \ GI = HI [orresponding angles] = y = 7 H y I z l and = + Since, l = 0 [lternate interior angles] lso, l = 0 [lternate interior angles] Now, = + = 0 + 0 = 30 8. The side of a is produced such that is on ray. The bisector of meets in L as in figure. rove that + = L. L Soln.: In, we have = + [terior angle property of a triangle] = +...(i) [Q L is the bisector of \ = ] In L, we have L = + L [terior anlge property of a triangle] L = + L = +...(ii) Subtracting (i) from (ii), we get L = + = L + = L. 9. bisects and =, prove that. Mathematics lass 9 LINS N NGLS 09

Soln.: In, we have = + [terior angle property of a triangle] = [Given = ] = [Q is bisected by ] = = [s alternate interior angles are equal] 0. In figure, lines Q and RS intersect each other at point, ray and ray bisect R and S respectively. If : = : 7, then find SQ and Q. R Soln.: R + S = 80 [Linear pair] We are given that, ray and ray bisect R and S respectively. Therefore, = R and = S. + = R + S ( ) = 80 = 90 Now, if : = : 7, then, we have = 90 = 0 and 9 = 7 90 = 70. 9 R = = 0 = 40 SQ = R [Vertically opposite angles] \ SQ = 40 Q = S + SQ = + SQ = 70 + 40 = 0 \ Q = 0. Q S = = S S. Ray bisects and is the ray opposite. Show that 3 = 4 3 4 Soln.: Ray and are opposite. So, is a straight line. \ 3 + = 80 (Linear pair)...(i) and 4 + = 80 (Linear pair)...(ii) rom (i) and (ii), we have 3 + = 4 + 3 + = 4 + (Q bisect = ) 3 = 4. In the figure, line l and m intersect at, forming angles as shown in the figure. l m z y v (i) If = 45, what is z? (ii) If v = 5, what is y? Soln.: (i) z = = 45 (Vertically opposite angles) (ii) y = v = 5 (Vertically opposite angles) 3. In the given figure, and GH. ind the values of, y, z and t. 00 R 50 H Q z t S Soln.: GH and RQ is a transversal y = 50 (orresponding angles) Q and RQ interesect at R. = 50 (Vertically opposite angles) Now, GH and Q is a transversal 00 + z = 80 (o-eterior angles) z = 80 00 z = 80 y G 0 LINS N NGLS lass 9 Mathematics

Q and QS is a transversal t = z (lternate interior angles) t = 80 4. In the figure,. If = 4 y 3 find the values of, y and z. y z and y z = 3 8 4 = 36 = 48 3 and, z= 8 y 3 8 = 36 z = 96 3 5. In the figure, prove that T QU. T U Soln.: and is a transversal ( + y) + z = 80 (co-interior angles) 4 3 5 3 y = 36 Now, = 4 y 3 8 + + = 80 3 4 3 8 Q = y and y = z z = y 3 8 3 = 80 = 80 3 5 Soln.: In QRS 85 Q 37 48 S = 48 + 37 (terior angle property = 85 UQR = 85 So, TR = UQR T UQ R of a triangle) (s corresponding angles are equal) Mathematics lass 9 LINS N NGLS

NRT STIN ercise 6.. In the given figure, lines and intersect at. If + = 70 and = 40, find and refle. Soln.: Since is a straight line, \ + + = 80 or ( + ) + = 80 or 70 + = 80 [Q + = 70 (Given)] or = 80 70 = 0 \ Refle = 360 0 = 50 lso, and intersect at. \ = [Vertically opposite angles] ut = 40 [Given] \ = 40 lso, + = 70 \ 40 + = 70 or = 70 40 = 30 Thus, = 30 and refle = 50.. In the given figure, lines XY and MN intersect at. If Y = 90 and a : b = : 3, find c. 90 b = 3 = 54 5 Since XY and MN intersect at, \ c = [a + Y] [Vertically opposite angles] or c = 36 + 90 = 6 Thus, the required measure of c = 6. 3. In the given figure, QR = RQ, then prove that QS = RT. S Q Soln.: ST is a straight line, \ QR + QS = 80...() [Linear pair] Similarly, RT + RQ = 80...() [Linear air] rom () and (), we have QS + QR = RT + RQ ut QR = RQ [Given] \ QS = RT 4. In the given figure, if + y = w + z, then prove that is a line. y z w R T Soln.: Since XY is a straight line. \ b + a + Y = 80 ut Y = 90 [Given] \ b + a = 80 90 = 90 lso a : b = : 3 90 a = + = 90 = 36 3 5 Soln.: Sum of all the angles at a point = 360 \ + y + z + w = 360 or, ( + y) + (z + w) = 360 ut ( + y) = (z + w) [Given] \( + y) + ( + y) = 360 or, ( + y) = 360 360 or, ( + y) = = 80 \ is a straight line. LINS N NGLS lass 9 Mathematics

5. In the given figure, Q is a line. Ray R is perpendicular to line Q. S is another ray lying between rays and R. rove that RS = QS S ( ). Q Soln.: Q is a straight line. S \ S + RS + RQ = 80 ut R ^ Q \ RQ = 90 S + RS + 90 = 80 S + RS = 90 R [Given] RS = 90 S... () Now, we have RS + RQ = QS RS + 90 = QS RS = QS 90... () dding () and (), we have RS = ( QS S) \ RS = QS S ( ). 6. It is given that XYZ = 64 and XY is produced to point. raw a figure from the given information. If ray YQ bisects ZY, find XYQ and refle QY. Soln.: XY is a straight line. Since XYQ = XYZ + ZYQ XYQ = 64 + QY [Q XYZ = 64 (Given) and ZYQ = QY] XYQ = 64 + 58 = [ QY = 58 ] Thus, XYQ = and refle QY = 30. ercise 6.. In the given figure, find the values of and y and then show that. 50 30 y Q Soln.: In the figure, we have and Q intersect at. \ y = 30...() [Vertically opposite angles] gain, Q is a straight line and stands on it. \ + Q = 80 [Linear pair] or 50 + = 80 = 80 50 = 30... () rom () and (), = y s they are pair of alternate interior angles. \. In the given figure, if, and y : z = 3 : 7, find. y \ XYZ + ZYQ + QY = 80 64 + ZYQ + QY = 80 [Q XYZ = 64 (given)] 64 + QY = 80 [YQ bisects ZY so, QY = ZYQ] QY = 80 64 = 6 6 QY = = 58 \ Refle QY = 360 58 = 30 Mathematics lass 9 Soln.: and [Given] \ and Q is a transversal. \ = z... () [lternate interior angles] gain, + y = 80 [o-interior ngles] z + y = 80 [y ()] z Q LINS N NGLS 3

ut y : z = 3 : 7 80 \ z = + = 80 7 7 = 6 ( 3 7) 0 rom () and (), we have = 6.... () 3. In the given figure, if, ^ and G = 6, find G, G and G. G Soln.: and G is a transversal. \ G = G [lternate interior angles] ut G = 6 [Given] \ G = 6 lso, G + = G or G + 90 = 6 [Q ^ (given)] G = 6 90 = 36 Now, and G is a transversal. \ G + G = 80 [o-interior angles] or G + 6 = 80 or G = 80 6 = 54 Thus, G = 6, G = 36 and G = 54. 4. In the given figure, if Q ST, QR = 0 and RST = 30, find QRS. [Hint: raw a line parallel to ST through point R.] Soln.: raw a line parallel to ST through R. 0 Q R S 30 Since Q ST [Given] and ST [onstruction] \ Q and QR is a transversal QR = QR [lternate Interior ngles] T ut QR = 0 [Given] \ QR = QRS + SR = 0...() gain ST and RS is a transversal \ RST + SR = 80 [o-interior angles] or 30 + SR = 80 SR = 80 30 = 50 Now, from (), we have QRS + 50 = 0 QRS = 0 50 = 60 Thus, QRS = 60. 5. In the given figure, if, Q = 50 and R = 7, find and y. Soln.: We have and Q is a transversal. \ Q = QR [lternate interior angles] or 50 = [Q Q = 50 (Given)] gain, and R is a transversal. \ R = R [lternate interior angles] R = 7 [Q R = 7 (given)] Q + QR = 7 50 + y = 7 [Q Q = 50 (given)] y = 7 50 = 77 Thus, = 50 and y = 77. 6. In the given figure, Q and RS are two mirrors placed parallel to each other. n incident ray strikes the mirror Q at, the reflected ray moves along the path and strikes the mirror RS at and again reflects back along. rove that. 4 LINS N NGLS lass 9 Mathematics

Soln.: raw ray L ^ Q and M ^ RS X 6 54 Q Q RS L M [Q L ^ Q and M ^ RS] Now, L M and is a transversal. \ L = M...() [lternate interior angles] Since, angle of incidence = ngle of reflection L = L and M = M L = M...() \ dding () and (), we have L + L = M + M = i.e., a pair of alternate interior angles are equal, \. ercise 6.3. In the adjoining figure, sides Q and RQ of QR are produced to points S and T respectively. If SR = 35 and QT = 0, find RQ. Y Soln.: In XYZ, we have XYZ + YZX + ZXY = 80 [ngle sum property of a triangle] ut XYZ = 54 and ZXY = 6 \ 54 + YZX + 6 = 80 YZX = 80 54 6 = 64 Y and Z are the bisectors of XYZ and XZY respectively, YZ= XYZ = 54 = 7 ( ) and ZY = YZX = 64 = 3 ( ) Now, in YZ, we have YZ + YZ + ZY = 80 [ngle sum property of a triangle] YZ + 7 + 3 = 80 YZ = 80 7 3 = Thus, ZY = 3 and YZ = 3. In the given figure, if, = 35 and = 53, find. 35 Z Soln.: We have, TQ + QR = 80 [Linear pair] 0 + QR = 80 QR = 80 0 = 70 Since, the side Q of QR is produced to S. QR + RQ = 35 [terior angle property of a ] 70 + RQ = 35 [ QR = 70 ] RQ = 35 70 RQ = 65. In the adjoining figure, X = 6, XYZ = 54. If Y and Z are the bisectors of XYZ and XZY respectively of XYZ, find ZY and YZ. Mathematics lass 9 53 Soln.: and is a transversal. So, = [lternate interior angles] and = 35 [Given] \ = 35 Now, in, we have + + = 80 [ngle sum property of a triangle] \ 53 + 35 + =80 [Q = = 35 and = 53 (Given)] = 80 53 35 = 9 Thus, = 9 LINS N NGLS 5

4. In the adjoining figure, if lines Q and RS intersect at point T, such that RT = 40, RT = 95 and TSQ = 75, find SQT. R 95 40 T 75 S Q Soln.: In RT, we have + R + TR = 80 [ngle sum property of a triangle] 95 + 40 + TR =80 [Q = 95, R = 40 (given)] TR =80 95 40 = 45 ut Q and RS intersect at T, \ TR = QTS [Vertically opposite angles] \ QTS = 45 [Q TR = 45 ] Now, in TQS, we have TSQ + STQ + SQT = 80 [ngle sum property of a triangle] \ 75 + 45 + SQT =80 [Q TSQ = 75 and STQ = 45 ] SQT = 80 75 45 = 60 Thus, SQT = 60 5. In the adjoining figure, if Q ^ S, Q SR, SQR = 8 and QRT = 65, then find the values of and y. Q 8 y 65 S R T Soln.: In QRS, the side SR is produced to T. \ QRT = RQS + RSQ [terior angle property of a triangle] ut RQS = 8 and QRT = 65 So, 8 + RSQ = 65 RSQ = 65 8 = 37 Since, Q SR and QS is a transversal. \ QS = RSQ = 37 [lternate interior angles] = 37 gain, Q ^ S = 90 Now, in QS, we have + QS + SQ = 80 [ngle sum property of a triangle] 90 + 37 + y = 80 y = 80 90 37 = 53 Thus, = 37 and y = 53 6. In the adjoining figure, the side QR of QR is produced to a point S. If the bisectors of QR and RS meet at point T, then prove that QTR = QR. Soln.: In QR, side QR is produced to S, so by eterior angle property, RS = + QR RS= + QR...() TRS= + TQR...() [Q QT and RT are bisectors of QR and RS respectively.] Now, in QRT, we have TRS = TQR + T...(3) [terior angle property of a triangle] rom () and (3), we have TQR+ = TQR+ T = T QR= QTR or QTR= QR 6 LINS N NGLS lass 9 Mathematics

XRIS Multiple hoice Questions Level-. n angle is 8 less than its complementary angle. The measure of this angle is (a) 36 (b) 48 (c) 83 (d) 8. Supplement of an angle is one fourth of itself. The measure of the angle is (a) 8 (b) 36 (c) 44 (d) 7 3. Line and intersect to. If = (3 0 ) and = (0 ), then the value of, is (a) 6 (b) (c) 36 (d) 30 4. If l m, then value of is 0 (a) 60 (b) 0 (c) 40 (d) annot be determined 5. The value of from the adjoining figure, if l m is (a) 5 (b) 0 (c) 9 (d) 36 6. In, : : = : 3 : 5, then angle at is (a) 54 (b) 6 (c) 36 (d) 64 7. In adjoining figure if = (3 + ), = ( 3 ), = 7, then = Mathematics lass 9 p l m (a) 4 (b) 3 (c) 96 (d) 98 8. The value of if is a straight line, is (a) 36 (b) 60 (c) 30 (d) 35 9. If, what is the value of? 5 (a) 8 (b) 5 (c) 0 (d) 5 0. If two parallel lines are intersected by a transversal, then each pair of corresponding angles so formed is (a) qual (b) omplementary (c) Supplementry (d) None of these. n angle is 4 more than its complementary angle, then angle is (a) 30 (b) 5 (c) 50 (d) None of these. If the supplement of an angle is three times its complement, then angle is (a) 40 (b) 35 (c) 50 (d) 45 3. Which one of the following statement is not false? (a) If two angles forming a linear pair, then each of these angle is of measure 90. (b) ngles forming a linear pair can both be acute angles. (c) oth of the angles forming a linear pair can be obtuse angles. (d) isectors of the adjacent angles forming a linear pair form a right angle. LINS N NGLS 7

4. alculate the value of. 8 (a) 70 (b) 70 (c) 5 (d) 45 3 5. alculate the value of. (a) 4 (b) 70 (c) 05 (d) 45 6. In figure, if l m, then = (a) 45 (b) 5 (c) 5 (d) 40 7. In Q.6, value of y = (a) 45 (b) 65 (c) 55 (d) 8 8. In figure, if l m, l n and : y = 3 :, then z = (a) 0 (b) 6 (c) 08 (d) 54 9. In figure, if l m, then = (a) 30 (b) 40 (c) 0 (d) 54. ind the value of. 30 5 (a) 70 (b) 75 (c) 60 (d) 65. ind. 50 75 (a) 45 (b) 60 (c) 50 (d) 55 3. If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 5 : 4, then the greater of the two angles is (a) 54 (b) 00 (c) 0 (d) 36 4. If is a straight line, then is 0 + 40 (a) 60 (b) 30 (c) 90 (d) 40 5. In the adjoining figure, if l m and n be the transversal, then the relation between and is n l m 70 (a) 0 (b) 0 (c) 90 (d) 98 0. In figure, if l m, m n, then = 50 l m l m n (a) = (b) + = 80 (c) = 90 (d) + = 90 Level- 6. If angle with measure and y form a complementary pair, then angles with which of the following measures will form a supplementary pair? (a) ( + 47 ), (y + 43 ) (b) ( 3 ), (y + 3 ) (c) ( 43 ), (y 47 ) (d) No such pair is possible 8 LINS N NGLS lass 9 Mathematics

7. If one angle of a triangle is equal to the sum of the other two angles, then triangle is a/an (a) acute angled triangle (b) obtuse angled triangle (c) right angled triangle (d) none of these 8. In figure, if, and y : z = 4 : 5, = (a) 00 (b) 76 (c) 8 (d) 9. In figure, lines and intersect at. If + = 00 and = 60, find and refle respectively. 3. In figure, and transversal cuts them at G and H respectively. If G = 0, then GH =.... 4. In figure, a transversal Q intersects two parallel lines and at L and M respectively. If = 95, then =.... L M 5. In figure, if is a straight line, then =.... Q 6. In figure, is a straight line and is greater than by 60, then =... (a) 40, 80 (b) 40, 60 (c) 30, 60 (d) 30, 50 30. In figure, lines XY and MN intersect at. If Y = 70 and : y = 3 :, find z. 7. In figure, if = 7 + 0 and = 3, the magnitude of which makes a straight line is.... (a) 70 (b) 95 (c) 36 (d) 0 ill in the lanks. If a transversal intersects two parallel lines, then the sum of the interior angles on the same side of the transversal is.... 8. In figure, is a straight line, if =, then =.... 9. In figure, =.... ( + 0) ( + 40). Lines which are parallel to the same line are... to each other. Mathematics lass 9 LINS N NGLS 9

0. In figure, =..., y =..., z =... True or alse. ngles forming a linear pair are supplementary.. If two adjacent angles are equal, then each angle measure 90. 3. ngles forming a linear pair can be both acute angles. 4. If angles forming a linear pair are equal, then each of these angles is of measure 90. 5. If two lines intersect each other and one pair of vertically opposite angles is formed by acute angles, then the other pair of vertically opposite angles will be formed by obtuse angles. 6. If two lines are intersected by a transversal, then corresponding angles are equal. 7. If two parallel lines are intersected by a transversal, then alternate angles are equal. 8. If two lines are intersected by a transversal, then angles on the same side of transversal are supplementary. 9. Two angles are complementary if their sum is 90. 0. The supplementary angles have their sum equal to 360. Match the ollowing In this section each question has two matching lists. hoices for the correct combination of elements from List-I and List-II are given as options (a), (b), (c) and (d) out of which one is correct.. Use the given figure to match List-I with List-II. z y 55 () List-I ngles m and y are List-II () lternate interior pair of angles (Q) ngles a and d are () lternate eterior pair of angles (R) ngles d and u are (3) Vertically opposite angles (S) ngles u and g are (4) orresponding angles ode : Q R S (a) 3 4 (b) 4 3 (c) 4 3 (d) 4 3. Use the given figure to match List-I with List-II., List-I List-II () orresponding () = 7 angles (Q) lternate interior () 4 + 5 = 80 angles (R) lternate eterior (3) = 5 angles (S) o-interior angles (4) 4 = 6 ode : Q R S (a) 4 3 (b) 3 4 (c) 4 3 (d) 3 4 ssertion & Reason Type irections : In each of the following questions, a statement of ssertion is given followed by a corresponding statement of Reason just below it. f the statements, mark the correct answer as (a) If both assertion and reason are true and reason is the correct eplanation of assertion. (b) If both assertion and reason are true but reason is not the correct eplanation of assertion. (c) If assertion is true but reason is false. (d) If assertion is false but reason is true. 0 LINS N NGLS lass 9 Mathematics

. ssertion : Two adjacent angles always form a linear pair. Reason : In a linear pair of angles two noncommon arms are opposite rays.. ssertion : The bisectors of the angles of a linear pair are at right angles. Reason : If the sum of two adjacent angles is 80, then the non-common arms of the angles are in a straight line. 3. ssertion : If a line is perpendicular to one of the two given parallel lines then it is also perpendicular to the other line. Reason : If two lines are intersected by a transversal then the bisectors of any pair of alternate interior angles are parallel. 4. ssertion : In figure, if XY is parallel to Q, then the angles and y are 70 and 45 respectively. (ii) each pair of alternate interior angles are equal (iii) each pair of co-interior angles are supplementary.. In the given figure, Q. The values of and y respectively are 75 5 y 0 G (a) 50, 70 (b) 70, 50 (c) 75, 45 (d) 0, 75. In the figure,. Then the value of p + q r = Q (a) 80 (b) 80 (c) 00 (d) 360 Reason : Sum of angles of a triangle is 80. 5. ssertion : In the given figure, if and = 30, then is 0. 3. In the given figure, Q RS and QXN = 05, RYN = 45, find XNY. Reason : If two parallel lines are intersected by a transversal, then co-interior angles are equal. (a) 45 (b) 30 (c) 60 (d) 0 SSG-II : The sum of all the angles formed on the same side of a line at a given point on the line is 80.. ind largest angle, if is a straight line. omprehension Type SSG-I : If a transversal intersects two parallel lines, then (i) each pair of corresponding angles are equal y y 3y (a) 60 (b) 30 (c) 80 (d) 90 Mathematics lass 9 LINS N NGLS

. If three straight lines, Q and RS intersect at. ind the value of from the given figure. (a) 0 (b) 7 (c) 7 (d) 54 3. In the given figure and are two lines intersecting at. If and are in ratio : 3, find all angles. (a) 7, 08, 08, 7 (b) 45, 00, 00, 45 (c) 30, 0, 30, 0 (d) 50, 0, 50, 0 SSG-III : If a transversal intersects two lines in such a way that a pair of alternate interior angles are equal, or each pair of co-interior angles are supplementary then the two lines are parallel.. rom the given figure for what value of if l m. (a) 60 (b) 90 (c) 50 (d) 0 Subjective roblems Very Short nswer Type. In the figure, find the angles marked as and y.. In the figure,. ind the values of and y. 3. In the figure and, find a and b. 0 l m (a) 0 (b) 80 (c) 60 (d) 00. In the given figure, is an isosceles triangle with = and, = 50. ind 50 (a) 50 (b) 60 (c) 65 (d) 70 4. In the figure, find the value of, if is bisector of angle in the triangle. 50 30 5. In figure, if QT ^ R, TQR = 40 and SR = 30, find. 30 T 3. In the given figure. If = 45 and = 75. ind. Q 40 S R LINS N NGLS lass 9 Mathematics

6. In figure,, = 70 and = 40, then find. 70 7. In figure, R and QR form a linear pair. If a b = 80, find the value of a and b. a R 40 Q 8. In the figure,. ind the value of. 00 b 0 9. rom the adjoining figure, find, y and z. 70 y z. In figure, and. lso ^. If = 55, find the values of, y and z. z 55 y 3. In the figure, and Q RS, find the value of angles,, 3, 4, 5, 6 and 7. 3 36 Q R 4 7 90 5 6 S 4. The sides and of the parallelogram are produced as shown in figure. rove that a + b = + y. y a 0. In the given figure,, Q. ind value of and y. b 5. In the figure given, and G find and y. 65 y 5 G H Short nswer Type. In the given figure, divides in the ratio : 3 and =. etermine the value of. 08 6. QRST is a regular pentagon and bisector of TQ meets SR at L. If bisector of SRQ meets L at M, find RML. S L T R M Q Mathematics lass 9 LINS N NGLS 3

7. In figure, if, G,, = 55 and 4 = 60, then find and 3. 4 8. In the adjoining figure bisects, bisects and ^. Show that,, are collinear. 9. In figure,. rove that + = 80 +. 3 0. In the given figure,. ind the value of. 05 5 G Long nswer Type. In the given figure, S is the angle bisector of the QR. T ^ QR. rove that TS = 7.5. 60 Q T S 45 R. In the given figure, b a = 65 and = 90, find the measure of, and. 3. In a, = 90 and ^. rove that =. 4. In figure, the sides and of are produced to points and respectively. If bisectors and of and respectively meet at point, then find. 5. In the figure, l m n and. ind the value of angles,, 3, 4, 5 and 6. 6 6 5 8 Integer nswer Type 4 3 In this section, each question, when worked out will result in one integer from 0 to 9 (both inclusive).. In the given figure,. ind the value of 5. 50 00. If an angle is supplement of itself. Then value of 60 is 6 6 l m n y 4 LINS N NGLS lass 9 Mathematics

3. ngles of a triangle are in ratio : : 3, then the greatest angle is what times to the smallest angle. 4. If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio of : 4. Then what will be the result if difference of the angles is divided by smaller angle. irections (5 0) : In the given figure, l m and p q. 5. a = 6. ngle b when divided by is 7. What is the sum of digits of angle c? 8. What is the sum of digits of angle d? 9. Value of e + 5 is 0 0. alculate f 5. 5 Mathematics lass 9 LINS N NGLS 5

HTR 6 Lines and ngles Multiple hoice Questions. (a) : Let the angle be. \ its complement = + 8 + + 8 = 90 = 90 8 = 7 = 36. (c) : Let the angle be \ its supplement = of = 4 4 5 + = 80 = 80 4 4 80 4 = = 44 5 3. (a) : Since vertically opposite angles are always equal \ (3 0 ) = (0 ) 3 + = 0 + 0 5 = 30 = 6. 4. (a) : + 0 = 80 [Linear pair] = 80 0 = 60 Since l m = = 60 M36 LINS N NGLS [orresponding ngles] 5. (a) : Since l m 0 + 5 = 80 [o-interior angles] 0 + 4 = 80 4 = 60 = 5 6. (a) : : : = : 3 : 5 =, = 3, = 5 \ + + = + 3 + 5 = 0 0 = 80 [ngle sum property of a triangle] = 8 = 3 8 = 54 7. (d): In = + [y eterior angle property of a triangle] 7 = 3 + + 3 7 = 4 8 = 4 = 3 \ = (3 + ) = 3 3 + = 96 + = 98 8. (b): = [Vertically opposite angles] Since is a straight line + + = 80 3 = 80 \ = 60 9. (c) : Since + + + 5 = 80 [o-interior angles] 9 = 80 \ = 0 0. (a). (b): Let the angle be. omplement of = (90 ) Since the difference is 4, we have (90 ) = 4 = 04 = 5.. (d): Let the angle be. omplement of = 90 Supplement of = 80 Given that, 80 = 3 (90 ) 80 = 70 3 = 70 80 = 90 = 45. lass 9 Mathematics

3. (d) 4. (c) : 8 + 3 + = 80 [Linear air] = 80 = 5. 5. (a) : 04 + 90 + 5 + = 360 [omplete ngle] = 4. 6. (b): Since l m, then y = 55 [orresponding ngles] Now, + y = 80 [Linear air] + 55 = 80 = 5 7. (c) 8. (c) : We have, l m, l n m n. Now, : y = 3 : = 3 y Mathematics lass 9... (i) lso + y = 80 [o-interior ngles] 3 + = 80 = 7 lso, = z...(ii) [lternate Interior ngles] rom (i) and (ii), we have 3 3 z= y z = 7 = 08 9. (b) : Since l m, 70 + = 80 [o-interior ngles] = 0 0. (a) : Since l m and m n, then l n + 50 = 80 [o-interior ngles] = 30.. (b) : + (80 30 ) + (80 5 ) = 80 [ngle sum property of a triangle] + 50 + 55 = 80 = 75.. (d) : = 80 (50 + 75 ) = 55 [ngle sum property of a triangle] \ = = 55 [Vertically opposite angles] 3. (b): Let the angle be 5 and 4. Since, these two angles are co-interior angles. So, we have 5 + 4 = 80 9 = 80 = 0. Hence, greater angle = 5 = 5 0 = 00 4. (d): Since, is a straight line \ ( + 40 ) + ( 0 ) + = 80 4 + 0 = 80 4 = 60 = 40 5. (b): Since l m and n is a transversal line. \ = (lternate interior angles) and + = 80 (Linear pair) + = 80 6. (a) : and y forms a complementary pair + y = 90 Now, + 47 + y + 43 = + y + 47 + 43 = + y + 90 = 90 + 90 = 80 \ ( + 47 ) and (y + 43 ) form a supplementary pair. 7. (c) : Let,, be the three angles of a triangle. = + [Given] lso, + + = 80 [ngle sum property of a triangle] + = 80 = 80 = 90 \ Triangle is a right angled triangle. 8. (a) : Given that and \ + y = 80...(i) [o-interior ngles] y + z = 80...(ii) [ HI = y, (Vertically pposite ngles)] Given that y : z = 4 : 5 y = 4 z 5 Substituting in (ii), we get 4 5 9 + = 80 = 80 5 z = 80 5 z = 00 9 Substituting value of z in (ii), we get y + 00 = 80 y = 80 rom (i), we have + 80 = 80 = 00. LINS N NGLS M37

9. (a) : and intersect at. = (vertically opposite angles) = 60 Now, + = 00 (given) = 00 60 = 40 is a straight line. + + = 80 60 + + 40 = 80 = 80 00 = 80 Refle = 360 80 = 80 30. (c) : : y = 3 : Let = 3a and y = a Now, XY is a straight line. \ + y + Y = 80 3a + a + 70 = 80 5a = 0 a = Hence, = 3a = 3 = 66 and y = a = = 44 Now, XY and MN intersect at. \ z = + Y (vertically opposite angles) z = 66 + 70 z = 36 ill in the lanks. 80. arallel 3. 70 : and G = 0 GH = G = 0 [Vertically pposite ngles] nd, GH + GH = 80 [o-interior ngles] GH = 80 0 = 70 4. 95 : = LM [Vertically pposite ngles] LM = 95 \ = 95 [orresponding ngles] 5. 43 : is straight line () + ( + 3) + ( + 5) = 80 4 = 80 8 4 = 7 = 43 6. 0 : = + 60 is a straight line \ + = 80 = 80 60 = 60 So, = 0 7. 6 : or to be a straight line 8. + = 80 7 + 0 + 3 = 80 0 = 60 \ = 6 80 : is a straight line. 3 + + = 80 ( + 40) + ( + 0) + ( + 40) = 80 3 + 00 = 80 3 = 80 \ = 80 3 9. 59 : + 4 + = 80 [Linear air] 3 + 3 = 80 3 = 77 = 59 0. 55, 5, 5 : = 55 lso, + y = 80 [Vertically pposite ngles] y = 80 55 = 5 [Linear air] Now, z = y = 5 [Vertically pposite \ = 55, y = 5, z = 5. True or alse. True ngles]. alse : The measure of each adjacent angle that are equal may not be equal to 90. 3. alse : ngles forming a linear pair cannot be both acute angles because then their sum cannot be 80. 4. True : + = 80 [Linear pair] = 80 = 90 \ If each angle forming a linear pair are equal then each angle is of measure 90. M38 LINS N NGLS lass 9 Mathematics

5. True : + = 80 [Linear pair] If < 90 > 90 3 4 cute ngle a (acute) (obtuse) and 3 are acute and and 4 are obtuse. 6. alse : If two parallel lines are intersected by a transversal, then corresponding angles are equal. 7. True 8. alse : Two lines must be parallel. 9. True 0. alse : Sum of supplementary angles is 80. Match the ollowing. (b): ; Q ; R 4; S 3 () ngles m and y lternate eterior pair of angles (Q) ngles a and d lternate interior pair of angles (R) ngles d and u orresponding angles (S) ngles u and g Vertically opposite angles.. (d): 3; Q 4 ; R ; S () orresponding angles = 5 (Q) lternate interior angles 4 = 6 (R) ltenate eterior angles = 7 (S) o-interior angles 4 + 5 = 80 ssertion & Reason Type. (d): Two adjacent angles do not always form a linear pair. In a linear pair of angles two non-common arms are opposite rays. ssertion: alse; Reason: True. (b): + = 80 [Linear air] 80 ( + ) = + = 90 + = 90. \ The bisectors of the angles of a linear pair are at right angles. ssertion: True; Reason: True but is not the correct eplanation of assertion. 3. (c) : l m and n ^ l = 90 [orresponding angles] n ^ m Reason is false. It can be stated in case of parallel lines. ssertion: True; Reason : alse. 4. (d): XY Q and is a transversal Q = Y (orresponding angles) 0 + y = 60 y = 60 0 y = 40 Now, in, + 0 = (ternal angle property) + 0 = 60 + 35 = 95 0 = 75 ssertion: alse; Reason: True n 90 l m Mathematics lass 9 LINS N NGLS M39

5. (c) : In, by pythagoras theorem. + + = 80 30 + 90 + = 80 = 80 0 = 60 and intersect at \ = (vertically opposite angles) = 60 Now, and is a transversal + = 80 (o-interior angles) = 80 60 = 0 ssertion: True; Reason: alse. omprehension Type SSG-I. (b): = Q [orresponding angles] 5 + y = 75 y = 50 M40 LINS N NGLS 75 5 y 0 G lso, + Q = 80 [o-interior angles] ( G + G) + (5 + y ) = 80 G + 0 + 5 + 50 = 80 G = 85 In G, G + G + = 80 85 + 5 + = 80 = 70.. (b): raw HK parallel to, let H be and H p G HG be y such that + y = q. p + = 80 [o-interior ngles] y = r [lternate Interior ngles] dding both equations, we get p + + y = 80 + r p + q = 80 + r p + q r = 80 r y Q K 3. (d) : onstruction : raw l Q + QXN = 80 [o-interior ngles] \ = 80 05 = 75 nd, \ = RYN [lternate Interior ngles] = 45 XNY = + = 75 + 45 = 0 SSG-II. (d): y + y + 3y = 80 [Linear air] 6y = 80 y = 30 \ largest angle = 3y = 3 30 = 90. (c) : R =, = 4 and = RQ = 45 Q is a straight line R + + RQ = 80 [Linear air] + 4 + 45 = 80 5 = 80 45 = 35 = 7 3. (a) : + = 80 [Linear air] Let = and = 3 \ + 3 = 80 = 36 Thus, = 7, = 08, = 08, = 7. SSG-III. (c) : = 0 [Vertically pposite ngles] or l m, co-interior angles must be supplementary, so + = 80 0 + = 80 \ = 60. (c) : Since is an isosceles triangle. \ = = Now, in, + + = 80 [ngle sum property of ] = 80 50 = 30 = 65 Since = [lternate Interior ngles] \ = 65 + + = 80 [Linear air] \ = 80 65 50 = 65. lass 9 Mathematics

3. (d): [Given] = [lternate Interior ngles] = 45 Now, in, is eterior angle So, = + [terior angle property of a ] = 75 + 45 = 0 Subjective roblems Very Short nswer Type. + 5 = 80 [o-interior ngles] \ = 80 5 = 55 Now y + 5 = 75 [lternate Interior ngles] \ y = 75 5 = 50. Hence, = 55, y = 50.. G = [orresponding ngles] \ y + 0 = 58 y = 58 0 y = 38 gain, y + = [terior angle of G] \ 38 + = 58 + = 80 38 = 4 Hence, = 4, y = 38 3. = 55 and = b b = 55 [orresponding ngles] In, = 80 (60 + b ) [ngle sum property of a triangle] = 80 (60 + 55 ) = 80 5 = 65 \ a = = 65 [orresponding ngles] 4. In, = 80 (50 + 30 ) = 00 [ngle sum property of a triangle] In, = 80 (90 + 50 ) = 40 00 and = = = 50 Now, = = = 50 40 = 0 Hence, = 0 5. In TQR, we have TQR + QRT + RTQ = 80 [ngle sum property of ] 40 + 90 + = 80 30 + = 80 = 50 6., = 70, = 40 Since = [lternate interior angles] = 70 lso from = + [terior angle property of a ] = 70 + 40 = 0 7.... R and QR form a linear pair \ R + QR = 80 or a + b = 80...() ut a b = 80...() [Given] dding eq. () and (), we get 60 a = 60 = = 30 Substituting the value of a in (), we get 30 + b = 80 b = 80 30 = 50 \ a = 30, b = 50 8. raw Q parallel to and. a + 00 = 80 [o-interior angles] a = 80 Now, b + 0 = 80 [o-interior ngles] b = 60 0 0 a b Q Since, Q is a straight line a + + b = 80 80 + + 60 = 80 = 40. 9. Here = = 70 [orresponding angles as ] and = 90 Now, = = 70 [ = ] In, = 80 = 80 70 70 = 40 nd = = 40 [lternate angles] y = 40 Since, z + = 80 [o-interior angles] z = 80 90 = 90 z = 90. 0. Q = 70 [orresponding angles] lso S = Q = 00 [Vertically opposite angles] s Q + Q = 80 [o-interior angles] 00 + 70 + y = 80 y = 80 70 = 0 \ = 70 and y = 0 Mathematics lass 9 LINS N NGLS M4

M4 LINS N NGLS Short nswer Type. = (80 08 ) = 7 ivide 7 in the ratio : 3 we get 8 and 54. \ = 8 and = 54 lso, = = = 8, etc. Now, in + = 8 + = 08 = 90. y + 55 = 80 [o-interior ngles] y = 80 55 = 5 gain = y [, orresponding ngles] Therefore = 5 Now, since and, therefore,. So, + = 80 [o-interior ngle] Therefore, 90 + z + 55 = 80 z = 35. 3. = 90 [o-interior ngles, Q RS] 3 = 36 [Vertically pposite ngles] + 3 = 90 [, and 3 from a linear pair] = 90 36 = 54 7 = 90 [Linear air] 4 + 7 = + 36 [lternate Interior ngles] 4 = 36 5 = 80 36 [Linear air] 5 = 44 6 = 4 = 36 [Vertically pposite ngles] Hence, = 90 = 7, = 54, 3 = 4 = 6 = 36, 5 = 44. 4. Since, \ a =...() [lternate interior angles] and b = y...() [lternate interior angles] dding () and (), we get a + b = + y. 5. = = 65...(i) [lternate Interior ngles] y = 80 5 [Linear air] \ y = 55 = y = 55 [orresponding ngles] In, we have + + = 80 [ngle sum property of a triangle] \ = 80 (65 + 55 ) \ = 60 and = = 60 [orresponding ngles] Hence, = 60, y = 55. 6. QRST is a regular pentagon \ = Q = R = S = T = 540 = 08..(i) 5 L is the bisector of TQ QL = 08 = 54...(ii) In quadrilateral QRL, QL + QR + QRL + RL = 360 [ngle sum property of a quadrilateral] 54 + 08 + 08 + RL = 360 RL = 360 70 = 90 In LMR, MRL + RLM + RML = 80 [ngle sum property of a triangle] 54 + 90 + RML = 80 [MR is the bisector of SRQ] RML = 80 44 = 36. 7. ; G; = 55, 4 = 60 Now, 4 = 3 [lternate Interior ngles] 3 = 60 and G + 4 = 80 [o-interior ngles] = 80 60 = 0 = 0, 3 = 60. 8. Given : Ray bisects, i.e., =, Ray bisects. i.e., 3 = 4. lso ^ To rove : oints, and are collinear. roof : ^ = 90 + 3 = 90 Multiplying both sides by + 3 = 80 ( + ) + ( 3 + 3) = 80 ( + ) + ( 3 + 4) = 80 [ =, 4 = 3] + = 80 y the converse of Linear air iom Ray and Ray are opposite rays. is a line. Hence, the points, and are collinear. lass 9 Mathematics

9. Through, draw parallel to and. Now + = 80...() [o-interior angles] lso, =...() [lternate interior angles] dding () and (), we get \ + + = 80 + + = 80 + [... + = ] Hence proved. 0. rom, draw. 05 So, a + 90 + a + b + 5 + b = 360 [omplete angle] or, 3a + 3b + 05 = 360 or, 3a + 3b = 360 05 or, 3a + 3b = 55 or, a + b = 85...() dding () and (), we get 3b = 50 b = 50 rom (), we have 00 a = 65 a = 00 65 = 35 So, = a = 35 = b = 50 = 00 and = a + b + 5 = 35 + 50 + 5 = 70 + 50 + 5 = 35. 3. Given : with = 90 and ^. To rove : = roof : In, + + = 80 [ngle sum property of a triangle] 5 Now, and is the transversal. \ + = 80 [o-interior angles] + = 80 = (80 ) gain, and is the transversal. \ + = 80 [o-interior angles] 05 + 5 + (80 ) = 80 = 30. Hence, = 30. Mathematics lass 9 Long nswer Type. In QR, Q = 60, R = 45 \ QR = 80 ( Q + R) = 80 (60 + 45 ) = 80 05 = 75 s S is angle bisector of QR = QR 75 QS = = 37. 5 lso, TQ is a right angled triangle. So, QT = 80 (90 + 60 ) = 80 50 = 30 \ TS = QS QT = 37.5 30 = 7.5.. Here, we have b a = 65...() and = 90 + = 90...() [Q = 90 ] lso, in, ^ So, = 90 and + + = 80 [ngle sum property of a triangle] + = 90...() [Q = 90 ] rom () and (), we get + = + =. 4. In, + + y = 80... () [ngle sum property of a triangle] lso, 6 + y = [terior angle property of a triangle] + = 6 ( y) y 3 + =...() Similarly, 3 + =... (3) In, + + = 80 [ngle sum property of a triangle] LINS N NGLS M43

y + 3 + + 3 + = 80 [Using () and (3)] y + 6 + + = 80 + 6 80 6 + = 80 [using angle sum property of ] = 80 6 90 + 3 = 90 3 = 59 \ = 59 5. 6 + 6 = 80 [Linear air] \ 6 = 80 6 6 = 8 6 + 5 = 80 [o-interior ngles] 5 = 80 8 = 6 = 8 [lternate Interior ngle] 4 + 8 = + 6 [lternate angles as ] 4 + 8 = 8 + 8 \ 4 = 8 3 + 4 = 80 [Linear air] 3 = 80 8 = 6 + 8 = 80 [Linear air] = 80 8 = 5 Hence, = 8, = 5, 3 = 6, 4 = 8, 5 = 6, 6 = 8. Integer nswer Type. () : onstruction : Through draw a line GH. Now, G and is a transversal. \ G = = 50 [lternate interior angles] gain, H and is a transversal. \ H + = 80 [o-interior angles] H + 00 = 80 H = 80 Now, GH is a straight line. \ G + + H = 80 50 + + 80 = 80 = 50. 50 So, 5 = 5 =.. (5) : = 80 = 90 60 90 60 = = 5 6 6 3. (3) : Let angles be, and 3. So, + + 3 = 80 [ngle sum property of triangle] = 30 Smallest angle = 30 ; Greatest angle = 90 So, the greatest angle is 3 times to the smallest angle. 4. (3) : Let angles be and 4. + 4 = 80 5 = 80 = 36. ngles are 36 and 44 ifference ofangles 44 36 = = 3. Smaller angle 36 5. (9) : a + 30 + 85 = 360 [omplete angle] a = 45 a 45 = 5 5 = 9 6. (5) : b = 80 (30 + a) [s l m, co-interior angles] = 80 (30 + 45 ) b = 05 b 05 = = 5 7. (9) : Since a and c are alternate interior angles c = 45. [s p q] \ Sum of digits = 4 + 5 = 9 8. (3) : c + b + d = 80 [Linear pair] 45 + 05 + d = 80 d = 30 \ Sum of digits of d = 3 + 0 = 3. 9. (5) : c and e are vertically opposite angles. \ e = 45. So, e + 5 = 50 0 0 = 5 0. (4) : b and f are vertically opposite angles. \ f = 05 So, f 5 00 = 5 5 = 4. M44 LINS N NGLS lass 9 Mathematics