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Page No 177:

Question 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.

Answer:

Given: lmt 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 angles)3 +4 = 180°   (linear pair)or 4 = 180-3 = 180 - 70 = 110°



Page No 178:

Question 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.

Answer:



Given: lmt 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 = 5x = 5(15) = 75°and 2 = 7x = 7(15) = 105° 2+3 = 180°                 (linear pair)3 = 180-105 = 75°3+6 = 180                     (interior angles on the same side of the transversal are supplementary)6 = 180-3 = 105°and 6 =8 =105°            (vertically opposite angles) 1 = 75°      2 = 105°      3 = 75°      8 = 105° 

Page No 178:

Question 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.

Answer:

Given: lmt 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 = 180or 5x = 195or x = 391 = (2x-8) = (2×39-8) = 70°2  =(3x-7) = (3×39-7) = 110°

Page No 178:

Question 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.

Answer:

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 pair)or y° = 180° - 65° =115°or y= 115

Page No 178:

Question 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.

Answer:

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 when AC is taken as the transversal) x = 65 y  =45

Page No 178:

Question 6:

In the adjoining figure, it is given that CE || BA, ∠BAC = 80° and ∠ECD = 35°.
Find (i) ∠ACE, (ii) ∠ACB,  (iii) ∠ABC.

Answer:

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 = 180°orACB + 80°+35° =180°or ACB = 65°(iii) In ABC: BAC + ACB +ABC = 180°   (angle sum property)80° +65° + ABC  = 180°ABC  = 35°

Page No 178:

Question 7:

In the adjoining figure, it is being given that AO || CD, OB || CE and ∠AOB = 50°
Find the measure of ∠ECD.

Answer:

Given: AO CD             OB CE            AOB = 50°AOD = CDB = 50°              (when  AO CD and OB is the transversal)ECD +CDB = 180°             (consecutive interior angles are supplementary, DB CE and CD is the transversal)ECD = 180°-50° = 130°

Page No 178:

Question 8:

In the adjoining figure, it is given that AB || CD,AOB = 50° and ∠CDO = 40°.
Find the measure of ∠BOD.

Answer:

Given: ABCD            ABO = 50°            CDO = 40°Construction: Through O, draw EOFAB.ABO = BOF = 50°            (alternate angles, when ABEF and OB is a transversal)FOD = ODC = 40°          (alternate angles, when CDEF and OD is a transversal) BOD = BOF  + FODBOD = 50°+40° =90°

Page No 178:

Question 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.

Answer:

Given: AB CD              GL and HM are angle bisectors of AGH and GHD, respectively. AGH=GHD (alternate angles)or 12 AGH=12GHDor LGH = GHM    (given)Therefore, GL  HM as we know that if the angles of any pair of alternate interior angles are equal, then the lines are parallel. 



Page No 179:

Question 10:

In the given figure, AB || CD,
ABE = 120°, ∠ECD = 100° and ∠BEC = x°
Find the value of x.

Answer:

Given: AB CD            ABE = 120°           ECD = 100°           BEC = x°Construction: FEG ABNow, since ABFEG and ABCD, FEGCD EFGABCDABE = BEG = 120° (alternate angles)or x°+y° = 120°  ....(i)DCE = CEF = 100°   (alternate angles)or x°+z° = 100°   .....(ii)Also, x°+y°+z° = 180°       (FEG is a straight line)   ...(iii)Adding (i) and (ii):2x°+y°+z° = 220°or, x° +180° = 220° (substituting (iii))x° = 40° x = 40

Page No 179:

Question 11:

In the given figure, ABCD is a quadrilateral in which AB || DC and AD || BC.
Prove that ∠ADC = ∠ABC.

Answer:

Given: AB CD            AD BC1 + 2 = 180°            (ABCD and AD is the transversal)    ...(i)2+ 3 =180°              (ADBC and AB is the transversal)    ...(ii)From (i) and (ii):1 + 2 = 180° = 2+ 3 1 = 3ADC = ABC

Page No 179:

Question 12:

In the given figure, l || m and p || q.
Find the measure of each of the angles ∠a, ∠b, ∠c and ∠d.

Answer:

Given: lm pq 1 = 65°1 = a =  65°         (vertically opposite angles)a + d = 180°            (consecutive interior angles on the same side of a transversal are supplementary)or d = 180°-65°= 115°c+ d = 180°              (consecutive interior angles on the same side of a transversal are supplementary)or c = 180°-115°= 65°c+ b = 180°               (consecutive interior angles on the same side of a transversal are supplementary)or b = 180°-65°= 115° a = 65°b = 115°c= 65°d= 115°

Page No 179:

Question 13:

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

Answer:

Given: AB DCADBCBAC = 35°CAD =40° BAC = y=  35°              (alternate angles when ABDC)CAD = x = 40°                  (alternate angles when ADBC)  x = 40 y=  35

Page No 179:

Question 14:

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.

Answer:

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 =180-x = 180 -55 = 125y + x =180° (consecutive interior angles on the same side of transversal are supplementary)y  =180- x = 180- 55 =  125

Page No 179:

Question 15:

In each of the given figures, two lines l and m are cut by a transevrsal t.
Find whether l || m.

Answer:

(i) 1+2 = 180 (linear pair)or 130° + 2 = 180°or 2 = 50° 40° =3 lm(ii) 2+3 = 180° (linear pair)35°+3 = 180°3 = 145°= 145° = 1  lm(iii)2+3 = 180 (linear pair)3 =180°- 125° = 55°3 =55° 60° = 1 lm







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