RS Aggarwal 2018 Solutions for Class 7 Math Chapter 14 - Properties Of Parallel Lines

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RS Aggarwal 2018 Solutions for Class 7 Math Chapter 14 Properties Of Parallel Lines are provided here with simple step-by-step explanations. These solutions for Properties Of Parallel Lines are extremely popular among class 7 students for Math Properties Of Parallel Lines Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the RS Aggarwal 2018 Book of class 7 Math Chapter 14 are provided here for you for free. You will also love the ad-free experience on Meritnation’s RS Aggarwal 2018 Solutions. All RS Aggarwal 2018 Solutions for class 7 Math are prepared by experts and are 100% accurate.

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:

$Given : l ∥ m t is a transversal . ∠ 5 = 70 ° ∠ 5 = ∠ 3 = 70 ° (alternate interior angles) ∠ 5 + ∠ 8 = 180 ° (linear pair) or 70 ° + ∠ 8 = 180 ° ∠ 8 = 110 ° ∠ 1 = ∠ 3 = 70 ° (vertically opposite angles) ∠ 3 + ∠ 4 = 180 ° (linear pair) or ∠ 4 = 180 - ∠ 3 = 180 - 70 = 110 °$

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:

$Given : l ∥ m t is a transversal . ∠ 1 : ∠ 2 = 5 : 7 Let the angles measure 5 x and 7 x . ∠ 1 + ∠ 2 = 180 ° (linear pair) ∴ 5 x + 7 x = 180 or 12 x = 180 or x = 15 ∴ ∠ 1 = 5 x = 5 (15) = 75 ° and ∠ 2 = 7 x = 7 (15) = 105 ° ∠ 2 + ∠ 3 = 180 ° (linear pair) ∠ 3 = 180 - 105 = 75 ° ∠ 3 + ∠ 6 = 180 (interior angles on the same side of t h e transversal are supplementary) ∠ 6 = 180 - ∠ 3 = 105 ° and ∠ 6 = ∠ 8 = 105 ° (vertically opposite angles) ∴ ∠ 1 = 75 ° ∠ 2 = 105 ° ∠ 3 = 75 ° ∠ 8 = 105 °$

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:

$Given : l ∥ m t is a transversal . Let : ∠ 1 = (2 x - 8) ° ∠ 2 = (3 x - 7) ° We know that the consecutive interior angles are supplementary . ∴ ∠ 1 + ∠ 2 = 180 ° or (2 x - 8) + (3 x - 7) = 180 or 5 x - 15 = 180 or 5 x = 195 or x = 39 ∠ 1 = (2 x - 8) = (2 \times 39 - 8) = 70 ° ∠ 2 = (3 x - 7) = (3 \times 39 - 7) = 110 °$

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:

$Given : CE ∥ BA ∠ BAC = 80 °, ∠ ECD = 35 ° (i) ∠ BAC = ∠ ACE = 80 ° (alternate angles with AC as a transversal) (ii) ∠ ACB + ∠ ACD = 180 ° (linear pair) or ∠ ACB + ∠ ACE + ∠ ECD = 180 ° or ∠ ACB + 80 ° + 35 ° = 180 ° or ∠ ACB = 65 ° (iii) In ∆ ABC : ∠ BAC + ∠ ACB + ∠ ABC = 180 ° (angle sum property) 80 ° + 65 ° + ∠ ABC = 180 ° ∠ ABC = 35 °$

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:

$Given : AO ∥ CD OB ∥ CE ∠ AOB = 50 ° ∠ AOD = ∠ CDB = 50 ° (when AO ∥ CD and OB is the transversal) ∠ ECD + ∠ CDB = 180 ° (consecutive interior angles are supplementary, DB ∥ CE and C D is t h e transversal) ∠ ECD = 180 ° - 50 ° = 130 °$

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:

$Given : AB ∥ CD ∠ ABO = 50 ° ∠ CDO = 40 ° Construction : Through O, draw EOF ∥ AB . ∠ ABO = ∠ BOF = 50 ° (alternate angles, when AB ∥ EF and OB is a transversal) ∠ FOD = ∠ ODC = 40 ° (alternate angles, when CD ∥ EF and OD is a transversal) ∠ BOD = ∠ BOF + ∠ FOD ∠ BOD = 50 ° + 40 ° = 90 °$

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:

$Given : AB ∥ CD GL and HM are angle bisectors of ∠ AGH and ∠ GHD, respectively . ∠ AGH = ∠ GHD (alternate angles) or \frac{1}{2} ∠ AGH = \frac{1}{2} ∠ GHD or ∠ LGH = ∠ GHM (given) Therefore, GL ∥ HM as we know that if the angles of any pair of alternate interior angles are equal, then the lines are parallel .$

Page No 179:

Question 10:

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

Answer:

$Given : AB ∥ CD ∠ ABE = 120 ° ∠ ECD = 100 ° ∠ BEC = x ° Construction : FEG ∥ AB Now, \sin c e A B ∥ F E G and A B ∥ C D, F E G ∥ C D ∴ E F G ∥ A B ∥ C D ∠ ABE = ∠ BEG = 120 ° (alternate angles) or x ° + y ° = 120 ° . . . . (i) ∠ DCE = ∠ CEF = 100 ° (alternate angles) or x ° + z ° = 100 ° . . . . . (ii) Also, x ° + y ° + z ° = 180 ° (FEG is a s traight line) . . . (iii) Adding (i) and (ii) : 2 x ° + y ° + z ° = 220 ° or, x ° + 180 ° = 220 ° (substi tuting (iii)) x ° = 40 ° ∴ x = 40$

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:

$Given : AB ∥ CD AD ∥ BC ∠ 1 + ∠ 2 = 180 ° (AB ∥ CD and AD is the transversal) . . . (i) ∠ 2 + ∠ 3 = 180 ° (AD ∥ BC and AB is the transversal) . . . (ii) From (i) and (ii) : ∠ 1 + ∠ 2 = 180 ° = ∠ 2 + ∠ 3 ∠ 1 = ∠ 3 ∠ ADC = ∠ ABC$

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:

$Given : l ∥ m p ∥ q ∠ 1 = 65 ° ∴ ∠ 1 = ∠ a = 65 ° (vertically opposite angles) ∠ a + ∠ d = 180 ° (consecutive interior angles on the same side of a transversal are supplementary) or ∠ d = 180 ° - 65 ° = 115 ° ∠ c + ∠ d = 180 ° (consecutive interior angles on the same side of a transversal are supplementary) or ∠ c = 180 ° - 115 ° = 65 ° ∠ c + ∠ b = 180 ° (consecutive interior angles on the same side of a transversal are supplementary) or ∠ b = 180 ° - 65 ° = 115 ° ∴ ∠ a = 65 ° ∠ b = 115 ° ∠ c = 65 ° ∠ d = 115 °$

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:

$Given : AB ∥ DC AD ∥ BC ∠ BAC = 35 ° ∠ CAD = 40 ° ∴ ∠ BAC = y = 35 ° (alternate angles when AB ∥ DC) ∠ CAD = x = 40 ° (alternate angles when AD ∥ BC) ∴ x = 40 y = 35$

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:

$Given : AB ∥ CD ∠ BAE = 125 ° ∠ CAB + ∠ BAE = 180 ° or 125 ° + x ° = 180 ° or x = 55 x + z = 180 ° (c onsecutive interior angles on the same side of transversal are s upplementary) z = 180 - x = 180 - 55 = 125 y + x = 180 ° (c onsecutive interior angles on the same side of transversal are s upplementary) y = 180 - x = 180 - 55 = 125$

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:

$(i) ∠ 1 + ∠ 2 = 180 (linear pair) or 130^{°} + ∠ 2 = 180 ° or ∠ 2 = 50 ° \neq 40 ° = ∠ 3 ∴ l ∦ m (ii) ∠ 2 + ∠ 3 = 180 ° (linear pair) 35 ° + ∠ 3 = 180 ° ∠ 3 = 145 ° = 145 ° = ∠ 1 ∴ l ∥ m (iii) ∠ 2 + ∠ 3 = 180 (linear pair) ∠ 3 = 180 ° - 125 ° = 55 ° ∠ 3 = 55 ° \neq 60 ° = ∠ 1 ∴ l ∦ m$

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