RS Aggarwal Solutions: Properties of Parallel Lines | Mathematics (Maths) Class 7 PDF Download

 Page 1


Q u e s t i o n : 1
In the given figure, l || m and t is a transversal.
If ?5 = 70°, find the measure of each of the angles
?1, ?3, ?4 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?5 = 70° ?5 = ?3  = 70°    (alternate interior angles) ?5 + ?8 = 180°   (linear pair)or 70° + ?8 = 180° ?8 = 110° ?1 = ?3 = 70°   (vertically opposite
Q u e s t i o n : 2
In the given figure, l || m and t is a transversal. If ?1 and ?2 are in the ratio 5 : 7, find the measure of each of the angles
?1, ?2, ?3 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?1: ?2 = 5: 7Let the angles measure 5x and 7x.  ?1 + ?2 = 180°                   (linear pair) ? 5x + 7x = 180                   or 12x = 180or x = 15 ? ?1 = 5
Q u e s t i o n : 3
Two parallel lines l and m cut by a transversal t. If the interior angles of the same side of t be (2x - 8)° and (3x - 7)°, find the measure of each of these angles.
S o l u t i o n :
Given: l ? mt is a transversal. Let: ?1 = (2x -8)° ?2 = (3x -7)°We know that the consecutive interior angles are supplementary. ? ?1 + ?2 = 180°or  (2x -8) + (3x -7) = 180or 5x -15 =
Q u e s t i o n : 4
In the given figure, l || m. If s and t be transversals such that s is not parallel to t. find the values of x and y.
Page 2


Q u e s t i o n : 1
In the given figure, l || m and t is a transversal.
If ?5 = 70°, find the measure of each of the angles
?1, ?3, ?4 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?5 = 70° ?5 = ?3  = 70°    (alternate interior angles) ?5 + ?8 = 180°   (linear pair)or 70° + ?8 = 180° ?8 = 110° ?1 = ?3 = 70°   (vertically opposite
Q u e s t i o n : 2
In the given figure, l || m and t is a transversal. If ?1 and ?2 are in the ratio 5 : 7, find the measure of each of the angles
?1, ?2, ?3 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?1: ?2 = 5: 7Let the angles measure 5x and 7x.  ?1 + ?2 = 180°                   (linear pair) ? 5x + 7x = 180                   or 12x = 180or x = 15 ? ?1 = 5
Q u e s t i o n : 3
Two parallel lines l and m cut by a transversal t. If the interior angles of the same side of t be (2x - 8)° and (3x - 7)°, find the measure of each of these angles.
S o l u t i o n :
Given: l ? mt is a transversal. Let: ?1 = (2x -8)° ?2 = (3x -7)°We know that the consecutive interior angles are supplementary. ? ?1 + ?2 = 180°or  (2x -8) + (3x -7) = 180or 5x -15 =
Q u e s t i o n : 4
In the given figure, l || m. If s and t be transversals such that s is not parallel to t. find the values of x and y.
S o l u t i o n :
From the given figure:
? 1 = ?3 = 50° (corresponding angles)and ? 1 + x° = 180° (linear pair)or x°  = 180° - 50° = 130°or x = 130 ? 2 = ?4 = 65° (corresponding angles)and ? 2 + y° = 180° (linear
Q u e s t i o n : 5
In the given figure, ?B = 65° and ?C = 45° in ?ABC and DAE || BC. If ?DAB = x° and ?EAC = y°, find the values of x and y.
S o l u t i o n :
Given: ?B = 65° ?C = 45°DAE ? BC The given lines are parallel. ? x° = ?B = 65°      (alternate angles when AB is taken as the transversal) y° = ?C = 45°             (alternate angles
Q u e s t i o n : 6
In the adjoining figure, it is given that CE || BA, ?BAC = 80° and ?ECD = 35°.
Find i
?ACE, ii
?ACB,  iii
?ABC.
S o l u t i o n :
Given: CE ? BA ?BAC = 80°, ?ECD = 35°(i) ?BAC = ?ACE = 80°      (alternate angles with AC as a transversal)(ii) ?ACB + ?ACD = 180°    (linear pair)or ?ACB + ?ACE + ?ECD
Q u e s t i o n : 7
In the adjoining figure, it is being given that AO || CD, OB || CE and ?AOB = 50°
Find the measure of ?ECD.
S o l u t i o n :
Given: AO ? CD             OB ? CE            ?AOB = 50° ?AOD = ?CDB = 50°              (when  AO ? CD and OB is the transversal) ?ECD + ?CDB = 180°             (consecutive interior angles
Q u e s t i o n : 8
In the adjoining figure, it is given that AB || CD, ?AOB = 50° and ?CDO = 40°.
Page 3


Q u e s t i o n : 1
In the given figure, l || m and t is a transversal.
If ?5 = 70°, find the measure of each of the angles
?1, ?3, ?4 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?5 = 70° ?5 = ?3  = 70°    (alternate interior angles) ?5 + ?8 = 180°   (linear pair)or 70° + ?8 = 180° ?8 = 110° ?1 = ?3 = 70°   (vertically opposite
Q u e s t i o n : 2
In the given figure, l || m and t is a transversal. If ?1 and ?2 are in the ratio 5 : 7, find the measure of each of the angles
?1, ?2, ?3 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?1: ?2 = 5: 7Let the angles measure 5x and 7x.  ?1 + ?2 = 180°                   (linear pair) ? 5x + 7x = 180                   or 12x = 180or x = 15 ? ?1 = 5
Q u e s t i o n : 3
Two parallel lines l and m cut by a transversal t. If the interior angles of the same side of t be (2x - 8)° and (3x - 7)°, find the measure of each of these angles.
S o l u t i o n :
Given: l ? mt is a transversal. Let: ?1 = (2x -8)° ?2 = (3x -7)°We know that the consecutive interior angles are supplementary. ? ?1 + ?2 = 180°or  (2x -8) + (3x -7) = 180or 5x -15 =
Q u e s t i o n : 4
In the given figure, l || m. If s and t be transversals such that s is not parallel to t. find the values of x and y.
S o l u t i o n :
From the given figure:
? 1 = ?3 = 50° (corresponding angles)and ? 1 + x° = 180° (linear pair)or x°  = 180° - 50° = 130°or x = 130 ? 2 = ?4 = 65° (corresponding angles)and ? 2 + y° = 180° (linear
Q u e s t i o n : 5
In the given figure, ?B = 65° and ?C = 45° in ?ABC and DAE || BC. If ?DAB = x° and ?EAC = y°, find the values of x and y.
S o l u t i o n :
Given: ?B = 65° ?C = 45°DAE ? BC The given lines are parallel. ? x° = ?B = 65°      (alternate angles when AB is taken as the transversal) y° = ?C = 45°             (alternate angles
Q u e s t i o n : 6
In the adjoining figure, it is given that CE || BA, ?BAC = 80° and ?ECD = 35°.
Find i
?ACE, ii
?ACB,  iii
?ABC.
S o l u t i o n :
Given: CE ? BA ?BAC = 80°, ?ECD = 35°(i) ?BAC = ?ACE = 80°      (alternate angles with AC as a transversal)(ii) ?ACB + ?ACD = 180°    (linear pair)or ?ACB + ?ACE + ?ECD
Q u e s t i o n : 7
In the adjoining figure, it is being given that AO || CD, OB || CE and ?AOB = 50°
Find the measure of ?ECD.
S o l u t i o n :
Given: AO ? CD             OB ? CE            ?AOB = 50° ?AOD = ?CDB = 50°              (when  AO ? CD and OB is the transversal) ?ECD + ?CDB = 180°             (consecutive interior angles
Q u e s t i o n : 8
In the adjoining figure, it is given that AB || CD, ?AOB = 50° and ?CDO = 40°.
Find the measure of ?BOD.
S o l u t i o n :
Given: AB ? ?CD            ?ABO = 50°            ?CDO = 40°Construction: Through O, draw EOF ? AB. ?ABO = ?BOF = 50°            (alternate angles, when AB ? EF and OB is a transversal
Q u e s t i o n : 9
In the given figure, AB || CD and a transversal EF cuts them at G and H respectively.
If GL and HM are the bisectors of the alternate angles ?AGH and ?GHD respectively, prove that GL || HM.
S o l u t i o n :
Given: AB ? CD              GL and HM are angle bisectors of ?AGH and ?GHD, respectively. ?AGH = ?GHD alternate angles or 
1
2
 ?AGH =
1
2
?GHDor ?LGH = ?GHM    given Therefore
Q u e s t i o n : 1 0
In the given figure, AB || CD,
?ABE = 120°, ?ECD = 100° and ?BEC = x°
Find the value of x.
S o l u t i o n :
Given: AB ? CD            ?ABE = 120°           ?ECD = 100°           ?BEC = x°Construction: FEG ? ABNow, since AB ? FEG and AB ? CD, FEG ? CD ? EFG ? AB ? CD ?ABE = ?BEG
Q u e s t i o n : 1 1
In the given figure, ABCD is a quadrilateral in which AB || DC and AD || BC.
Prove that ?ADC = ?ABC.
S o l u t i o n :
Given: AB ? CD            AD ? BC ?1 + ?2 = 180°            (AB ? CD and AD is the transversal)    . . . (i) ?2 + ?3 = 180°              (AD ? BC and AB is the transversal)    . . . (ii)From (i) and (ii
( ) ( )
Page 4


Q u e s t i o n : 1
In the given figure, l || m and t is a transversal.
If ?5 = 70°, find the measure of each of the angles
?1, ?3, ?4 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?5 = 70° ?5 = ?3  = 70°    (alternate interior angles) ?5 + ?8 = 180°   (linear pair)or 70° + ?8 = 180° ?8 = 110° ?1 = ?3 = 70°   (vertically opposite
Q u e s t i o n : 2
In the given figure, l || m and t is a transversal. If ?1 and ?2 are in the ratio 5 : 7, find the measure of each of the angles
?1, ?2, ?3 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?1: ?2 = 5: 7Let the angles measure 5x and 7x.  ?1 + ?2 = 180°                   (linear pair) ? 5x + 7x = 180                   or 12x = 180or x = 15 ? ?1 = 5
Q u e s t i o n : 3
Two parallel lines l and m cut by a transversal t. If the interior angles of the same side of t be (2x - 8)° and (3x - 7)°, find the measure of each of these angles.
S o l u t i o n :
Given: l ? mt is a transversal. Let: ?1 = (2x -8)° ?2 = (3x -7)°We know that the consecutive interior angles are supplementary. ? ?1 + ?2 = 180°or  (2x -8) + (3x -7) = 180or 5x -15 =
Q u e s t i o n : 4
In the given figure, l || m. If s and t be transversals such that s is not parallel to t. find the values of x and y.
S o l u t i o n :
From the given figure:
? 1 = ?3 = 50° (corresponding angles)and ? 1 + x° = 180° (linear pair)or x°  = 180° - 50° = 130°or x = 130 ? 2 = ?4 = 65° (corresponding angles)and ? 2 + y° = 180° (linear
Q u e s t i o n : 5
In the given figure, ?B = 65° and ?C = 45° in ?ABC and DAE || BC. If ?DAB = x° and ?EAC = y°, find the values of x and y.
S o l u t i o n :
Given: ?B = 65° ?C = 45°DAE ? BC The given lines are parallel. ? x° = ?B = 65°      (alternate angles when AB is taken as the transversal) y° = ?C = 45°             (alternate angles
Q u e s t i o n : 6
In the adjoining figure, it is given that CE || BA, ?BAC = 80° and ?ECD = 35°.
Find i
?ACE, ii
?ACB,  iii
?ABC.
S o l u t i o n :
Given: CE ? BA ?BAC = 80°, ?ECD = 35°(i) ?BAC = ?ACE = 80°      (alternate angles with AC as a transversal)(ii) ?ACB + ?ACD = 180°    (linear pair)or ?ACB + ?ACE + ?ECD
Q u e s t i o n : 7
In the adjoining figure, it is being given that AO || CD, OB || CE and ?AOB = 50°
Find the measure of ?ECD.
S o l u t i o n :
Given: AO ? CD             OB ? CE            ?AOB = 50° ?AOD = ?CDB = 50°              (when  AO ? CD and OB is the transversal) ?ECD + ?CDB = 180°             (consecutive interior angles
Q u e s t i o n : 8
In the adjoining figure, it is given that AB || CD, ?AOB = 50° and ?CDO = 40°.
Find the measure of ?BOD.
S o l u t i o n :
Given: AB ? ?CD            ?ABO = 50°            ?CDO = 40°Construction: Through O, draw EOF ? AB. ?ABO = ?BOF = 50°            (alternate angles, when AB ? EF and OB is a transversal
Q u e s t i o n : 9
In the given figure, AB || CD and a transversal EF cuts them at G and H respectively.
If GL and HM are the bisectors of the alternate angles ?AGH and ?GHD respectively, prove that GL || HM.
S o l u t i o n :
Given: AB ? CD              GL and HM are angle bisectors of ?AGH and ?GHD, respectively. ?AGH = ?GHD alternate angles or 
1
2
 ?AGH =
1
2
?GHDor ?LGH = ?GHM    given Therefore
Q u e s t i o n : 1 0
In the given figure, AB || CD,
?ABE = 120°, ?ECD = 100° and ?BEC = x°
Find the value of x.
S o l u t i o n :
Given: AB ? CD            ?ABE = 120°           ?ECD = 100°           ?BEC = x°Construction: FEG ? ABNow, since AB ? FEG and AB ? CD, FEG ? CD ? EFG ? AB ? CD ?ABE = ?BEG
Q u e s t i o n : 1 1
In the given figure, ABCD is a quadrilateral in which AB || DC and AD || BC.
Prove that ?ADC = ?ABC.
S o l u t i o n :
Given: AB ? CD            AD ? BC ?1 + ?2 = 180°            (AB ? CD and AD is the transversal)    . . . (i) ?2 + ?3 = 180°              (AD ? BC and AB is the transversal)    . . . (ii)From (i) and (ii
( ) ( )
Q u e s t i o n : 1 2
In the given figure, l || m and p || q.
Find the measure of each of the angles ?a, ?b, ?c and ?d.
S o l u t i o n :
Given: l ? m p ? q ?1 = 65° ? ?1 = ?a =  65°         (vertically opposite angles) ?a + ?d = 180°            (consecutive interior angles on the same side of a transversal are supplementary
Q u e s t i o n : 1 3
In the given figure, AB || DC and AD || BC, and AC is a diagonal. If ?BAC = 35°, ?CAD = 40°, ?ACB = x° and ?ACD = y°, find the value of x and y
S o l u t i o n :
Given: AB ? DCAD ? BC ?BAC = 35° ?CAD = 40° ? ?BAC = y =  35°              (alternate angles when AB ? DC) ?CAD = x = 40°                  (alternate angles when AD ? BC) ?  x =
Q u e s t i o n : 1 4
In the given figure, AB || CD and CA has been produced to E so that ?BAE = 125°.
If ?BAC = x°, ?ABD = x°, ?BDC = y° and ?ACD = z°, find the values of x, y, z.
S o l u t i o n :
Given: AB ? CD ?BAE = 125° ?CAB + ?BAE  = 180°or 125° + x° = 180°or x = 55x + z = 180° (consecutive interior angles on the same side of transversal are supplementary)z =
Q u e s t i o n : 1 5
In each of the given figures, two lines l and m are cut by a transevrsal t.
Find whether l || m.
S o l u t i o n :
( ) ( ) ( ) ( ) ( )
Page 5


Q u e s t i o n : 1
In the given figure, l || m and t is a transversal.
If ?5 = 70°, find the measure of each of the angles
?1, ?3, ?4 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?5 = 70° ?5 = ?3  = 70°    (alternate interior angles) ?5 + ?8 = 180°   (linear pair)or 70° + ?8 = 180° ?8 = 110° ?1 = ?3 = 70°   (vertically opposite
Q u e s t i o n : 2
In the given figure, l || m and t is a transversal. If ?1 and ?2 are in the ratio 5 : 7, find the measure of each of the angles
?1, ?2, ?3 and ?8.
S o l u t i o n :
Given: l ? mt is a transversal. ?1: ?2 = 5: 7Let the angles measure 5x and 7x.  ?1 + ?2 = 180°                   (linear pair) ? 5x + 7x = 180                   or 12x = 180or x = 15 ? ?1 = 5
Q u e s t i o n : 3
Two parallel lines l and m cut by a transversal t. If the interior angles of the same side of t be (2x - 8)° and (3x - 7)°, find the measure of each of these angles.
S o l u t i o n :
Given: l ? mt is a transversal. Let: ?1 = (2x -8)° ?2 = (3x -7)°We know that the consecutive interior angles are supplementary. ? ?1 + ?2 = 180°or  (2x -8) + (3x -7) = 180or 5x -15 =
Q u e s t i o n : 4
In the given figure, l || m. If s and t be transversals such that s is not parallel to t. find the values of x and y.
S o l u t i o n :
From the given figure:
? 1 = ?3 = 50° (corresponding angles)and ? 1 + x° = 180° (linear pair)or x°  = 180° - 50° = 130°or x = 130 ? 2 = ?4 = 65° (corresponding angles)and ? 2 + y° = 180° (linear
Q u e s t i o n : 5
In the given figure, ?B = 65° and ?C = 45° in ?ABC and DAE || BC. If ?DAB = x° and ?EAC = y°, find the values of x and y.
S o l u t i o n :
Given: ?B = 65° ?C = 45°DAE ? BC The given lines are parallel. ? x° = ?B = 65°      (alternate angles when AB is taken as the transversal) y° = ?C = 45°             (alternate angles
Q u e s t i o n : 6
In the adjoining figure, it is given that CE || BA, ?BAC = 80° and ?ECD = 35°.
Find i
?ACE, ii
?ACB,  iii
?ABC.
S o l u t i o n :
Given: CE ? BA ?BAC = 80°, ?ECD = 35°(i) ?BAC = ?ACE = 80°      (alternate angles with AC as a transversal)(ii) ?ACB + ?ACD = 180°    (linear pair)or ?ACB + ?ACE + ?ECD
Q u e s t i o n : 7
In the adjoining figure, it is being given that AO || CD, OB || CE and ?AOB = 50°
Find the measure of ?ECD.
S o l u t i o n :
Given: AO ? CD             OB ? CE            ?AOB = 50° ?AOD = ?CDB = 50°              (when  AO ? CD and OB is the transversal) ?ECD + ?CDB = 180°             (consecutive interior angles
Q u e s t i o n : 8
In the adjoining figure, it is given that AB || CD, ?AOB = 50° and ?CDO = 40°.
Find the measure of ?BOD.
S o l u t i o n :
Given: AB ? ?CD            ?ABO = 50°            ?CDO = 40°Construction: Through O, draw EOF ? AB. ?ABO = ?BOF = 50°            (alternate angles, when AB ? EF and OB is a transversal
Q u e s t i o n : 9
In the given figure, AB || CD and a transversal EF cuts them at G and H respectively.
If GL and HM are the bisectors of the alternate angles ?AGH and ?GHD respectively, prove that GL || HM.
S o l u t i o n :
Given: AB ? CD              GL and HM are angle bisectors of ?AGH and ?GHD, respectively. ?AGH = ?GHD alternate angles or 
1
2
 ?AGH =
1
2
?GHDor ?LGH = ?GHM    given Therefore
Q u e s t i o n : 1 0
In the given figure, AB || CD,
?ABE = 120°, ?ECD = 100° and ?BEC = x°
Find the value of x.
S o l u t i o n :
Given: AB ? CD            ?ABE = 120°           ?ECD = 100°           ?BEC = x°Construction: FEG ? ABNow, since AB ? FEG and AB ? CD, FEG ? CD ? EFG ? AB ? CD ?ABE = ?BEG
Q u e s t i o n : 1 1
In the given figure, ABCD is a quadrilateral in which AB || DC and AD || BC.
Prove that ?ADC = ?ABC.
S o l u t i o n :
Given: AB ? CD            AD ? BC ?1 + ?2 = 180°            (AB ? CD and AD is the transversal)    . . . (i) ?2 + ?3 = 180°              (AD ? BC and AB is the transversal)    . . . (ii)From (i) and (ii
( ) ( )
Q u e s t i o n : 1 2
In the given figure, l || m and p || q.
Find the measure of each of the angles ?a, ?b, ?c and ?d.
S o l u t i o n :
Given: l ? m p ? q ?1 = 65° ? ?1 = ?a =  65°         (vertically opposite angles) ?a + ?d = 180°            (consecutive interior angles on the same side of a transversal are supplementary
Q u e s t i o n : 1 3
In the given figure, AB || DC and AD || BC, and AC is a diagonal. If ?BAC = 35°, ?CAD = 40°, ?ACB = x° and ?ACD = y°, find the value of x and y
S o l u t i o n :
Given: AB ? DCAD ? BC ?BAC = 35° ?CAD = 40° ? ?BAC = y =  35°              (alternate angles when AB ? DC) ?CAD = x = 40°                  (alternate angles when AD ? BC) ?  x =
Q u e s t i o n : 1 4
In the given figure, AB || CD and CA has been produced to E so that ?BAE = 125°.
If ?BAC = x°, ?ABD = x°, ?BDC = y° and ?ACD = z°, find the values of x, y, z.
S o l u t i o n :
Given: AB ? CD ?BAE = 125° ?CAB + ?BAE  = 180°or 125° + x° = 180°or x = 55x + z = 180° (consecutive interior angles on the same side of transversal are supplementary)z =
Q u e s t i o n : 1 5
In each of the given figures, two lines l and m are cut by a transevrsal t.
Find whether l || m.
S o l u t i o n :
( ) ( ) ( ) ( ) ( )
i ?1 + ?2 = 180 linear pair or 130
°
 + ?2 = 180°or ?2 = 50° ? 40° = ?3 ? l ? m ii ?2 + ?3 = 180° linear pair 35°+ ?3 = 180° ?3 = 145° = 145° = ?1 ? l ? m iii ?2 + ? ( ) ( ) ( ) ( ) ( )