Exercise 14
Page-177Question 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 1:
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 angles)∠3 +∠4 = 180° (linear pair)or ∠4 = 180-∠3 = 180 - 70 = 110°
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 2:
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 = 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°
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 3:
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 = 180or 5x = 195or x = 39∠1 = (2x-8) = (2×39-8) = 70°∠2 =(3x-7) = (3×39-7) = 110°
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 4:
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
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 5:
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
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 6:
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°or∠ACB + 80°+35° =180°or ∠ACB = 65°(iii) In ∆ABC: ∠BAC + ∠ACB +∠ABC = 180° (angle sum property)80° +65° + ∠ABC = 180°∠ABC = 35°
Question 7:
In the adjoining figure, it is being given that AO || CD, OB || CE and ∠AOB = 50°
Find the measure of ∠ECD.
Answer 7:
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°
Question 8:
In the adjoining figure, it is given that AB || CD, ∠AOB = 50° and ∠CDO = 40°.
Find the measure of ∠BOD.
Answer 8:
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)∠FOD = ∠ODC = 40° (alternate angles, when CD∥EF and OD is a transversal) ∠BOD = ∠BOF + ∠FOD∠BOD = 50°+40° =90°
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 9:
Given: AB ∥CD GL and HM are angle bisectors of ∠AGH and ∠GHD, respectively. ∠AGH=∠GHD (alternate angles)or 12 ∠AGH=12∠GHDor ∠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.
Question 10:
In the given figure, AB || CD,
∠ABE = 120°, ∠ECD = 100° and ∠BEC = x°
Find the value of x.
Answer 10:
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 = 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
Question 11:
In the given figure, ABCD is a quadrilateral in which AB || DC and AD || BC.
Prove that ∠ADC = ∠ABC.
Answer 11:
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):∠1 + ∠2 = 180° = ∠2+ ∠3 ∠1 = ∠3∠ADC = ∠ABC
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 12:
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)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°
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 13:
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 = 40 y= 35
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 14:
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
Question 15:
In each of the given figures, two lines l and m are cut by a transevrsal t.
Find whether l || m.
Answer 15:
(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+∠3 = 180 (linear pair)∠3 =180°- 125° = 55°∠3 =55°≠ 60° = ∠1∴ l∦m
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