Grade 9 Lines and Angles

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ID : cn-9-lines-and-angles [1] Grade 9 Lines and Angles For more such worksheets visit www.edugain.com Answer the questions (1) If AB and CD are parallel, find the value of x.

1 ID : cn-9-lines-and-angles [1] Grade 9 Lines and Angles For more such worksheets visit Answer the questions (1) If AB and CD are parallel, find the value of x. (2) Lines AB and CD intersect at point O. If AOC + BOE = 90 and BOD = 40, find BOE. (3) What is the value of the supplement of the complement of 66? (4) If AD and BD are bisectors of CAB and CBA, respectively. Find the sum of angles x and y.

2 (5) If the two horizontal lines given below are parallel, find the measure of angle x. ID : cn-9-lines-and-angles [2] (6) If AB and CD are parallel, find the measure of angle x. (7) If ADB is a right angle, find the value of angle x. (8) If AC and EF are parallel, find ADB. (9) Find the measure of a+b.

3 (10) If OD is perpendicular to AB, and DOC = 20, find BOC - AOC. ID : cn-9-lines-and-angles [3] (11) Find the measure of angle x. (12) If lines XY and MN intersect as shown below and a:b = 2:3, find c. (13) If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio 1:2, then find the smaller of the two angles. Choose correct answer(s) from the given choices (14) If the angles of a triangle are in the ratio 5:2:3, the triangle is a. an isosceles triangle b. an acute-angled triangle c. an obtuse-angled triangle d. a right-angled triangle (15) Which of the following can be true for a triangle? a. Each angle is less than 60 b. Two angles are obtuse angles c. Each angle is greater than 60 d. Each angle is equal to Edugain ( All Rights Reserved Many more such worksheets can be generated at

4 Answers ID : cn-9-lines-and-angles [4] (1) 65 It is given that lines AB and CD are parallel and the third line (say EF) cuts the lines AB and CD at a certain angle as shown in the figure above. Let us redraw the figure as below: a = c (Vertically opposite angles) c = e (Alternate interior angles) Therefore, we can write, a = c = e = g Again, b = d (Vertically opposite angles) d = f (Alternate interior angles) Therefore, we can write, b = d = f = h We know that sum of two adjacent angles is equal to 180. Therefore, we can write, a + b = 180, b + c = 180, c + d = 180, d + a = 180 Here, e = 115 and d = x e + h = h = 180 h = h = 65 As h is equal to d, x is 65.

5 Hence, the value of x is 65. ID : cn-9-lines-and-angles [5] (2) 50 We are given that BOD = 40. Now, look at the figure. BOD and AOC are vertically opposite angles. So, they are equal. This means, AOC = 40. Since, AOC + BOE = 90. From step 1, we have AOC = 40. We can say that 40 + BOE = 90. Now, subtracting 40 from both the sides, we get, BOE = 90-40, or BOE = 50 (3) 156 If we look at the question carefully, we notice that we have to find the complement of 66 and then find the supplement of the complement of 66. We know, the sum of the complementary angles is 90. Therefore, the complement of 66, = = 24 The sum of the supplementary angles is 180. Therefore, the supplement of 24, = = 156 Step 4 Hence, the value of the supplement of the complement of 66 is 156.

6 (4) 45 ID : cn-9-lines-and-angles [6] It is given that AD and BD are bisectors of CAB and CBA, respectively. Therefore, x = CAB/ (1) y = CBA/ (2) In triangle ABC, CAB + CBA + ACB = (The sum of all the angles of a triangle is 180 ) CAB + CBA + 90 = 180 CAB + CBA = CAB + CBA = 90 CAB/2 + CBA/2 = 90/2 x + y = 45...(From equations (1) and (2)) Hence, the sum of the angles x and y is 45. (5) 60 When a straight line cuts any two parallel lines, its corresponding angles are equal. Since, angles x and 60 are corresponding angles. We get, x = 60 (corresponding angles of two parallel lines are equal)

7 (6) 26 ID : cn-9-lines-and-angles [7] The parallel lines AB and CD are intersected by a transversal as shown below, Here, P and Q are consecutive interior angles, i.e., P + Q = 180 On comparing the given angles with P and Q, 4x + 1x + 50 = 180 5x = 130 Hence, x = 26.

8 (7) 86 ID : cn-9-lines-and-angles [8] According to the question ADB is the right angle, ADB = 90, DAB = 30 In right angled triangle ΔADB, ADB + DAB + ABD = 180 [Since, the sum of all the angles of a triangle is equal to 180 ] ABD = ABD = 180 ABD = 60 In ΔBEC, ABD + BEC + 26 = BEC + 26 = 180 BEC + 86 = 180 BEC = BEC = 94 x + BEC = 180 [Since, the sum of all angles on one side of a straight line is equal to 180.] x = BEC x = x = 86 Step 4 Hence, the value of angle x is 86.

9 (8) 90 ID : cn-9-lines-and-angles [9] If we look at the figure carefully, we notice that ADE = 45 and DBC = 135. According to the question, AC and EF are parallel. Therefore, we can say that EDB and DBC are alternate interior angles. Now, EDB = DBC (Alternate interior angles) ADE + ADB = DBC (As, EDB = ADE + ADB) ADB = ADE (As, ADB = 135 ) ADB = (As, ADE = 45 ) ADB = 90 Step 4 Therefore, ADB = 90. (9) 270 If we look at the figure carefully, we notice that the angles b, 90, and a are made around a point. The sum of all the angles around a point is 360. Therefore, b a = 360 a + b = a + b = 270 Hence, the measure of a + b is 270.

10 (10) 40 ID : cn-9-lines-and-angles [10] According to the question, DOC = 20 and OD is perpendicular to AB. Therefore, AOD = 90 and BOD = 90. Also, DOC + AOC = AOD 20 + AOC = 90 (As, AOD = 90 and DOC = 20 ) AOC = AOC = 70 Now, BOC - AOC = BOD + DOC - AOC (As, BOC = BOD + DOC) = (As, BOD = 90, DOC = 20 and AOC = 70 ) = 40 Step 4 Therefore, BOC - AOC = 40 (11) 70 If we look at the figure carefully, we notice that line AB is a straight line. We know, the angles on a straight line add up to 180. So, x = x = 180 x = x = 70 Hence, the measure of angle x is 70.

11 (12) 126 ID : cn-9-lines-and-angles [11] As XY is a straight line, and the sum of the angles on a straight line is equal to 180. We have, a + b + 90 = 180 a + b = 90. We may notice that the angles MOY and XON are vertically opposite angles. So, a + 90 = c (vertically opposite angles are equal). We are given a:b = 2:3, or a b = 2 3. Cross multiplying the fractions, we get, 3a = 2b. Step 4 We put b = 3a 2 in a + b = 90 and get, a + 3a 2 = 90, or 5a 2 = 90, or 5a = 180. Dividing each side by 5, we get, a = 180, or Step 5 Now, since b = 3a 2. We can say that b = , or b = 54. Step 6 From step 2, we have, c = a + 90, or c = , or c = 126. (13) 60 Let x be the first interior angle on the same side of a transversal intersecting two parallel lines.

12 ID : cn-9-lines-and-angles [12] We know that the sum of two interior angles on the same side of a transversal intersecting two parallel lines is 180. Thus, the second interior angle on the same side of a transversal intersecting two parallel lines = x The ratio of the two interior angles on the same side of a transversal intersecting two parallel lines x = x It is given that the ratio of the two interior angles on the same side of a transversal intersecting two parallel lines = 1:2 x Therefore, = x 2 By cross multiplying, we get: 2x = 1(180 - x) 2x = x 2x + 1x = 180 3x = 180 x = x = 60 Step 4 The first angle = 60 The second angle = = 120 Step 5 Hence, the smaller of the two angles is 60.

13 (14) d. a right-angled triangle ID : cn-9-lines-and-angles [13] According to the question, all the angles of the triangle are in the ratio 5:2:3. We can assume the three angles of the triangle to be 5x, 2x, and 3x where x is a common factor. We know that the sum of the three angles of a triangle is 180. Therefore, 5x + 2x + 3x = x = 180 x = Now, 5x = = 90, 2x = = 36, and 3x = = Therefore, the three angles of the triangle are 90, 36, and 54. Since, one of the angles of the triangle is 90, the triangle is a right-angled triangle. (15) d. Each angle is equal to 60 The sum of all the angles of a triangle must be 180. If we look at the all of the options carefully, we notice that the statement "Each angle is equal to 60 " satisfies the condition of a triangle. Hence, the statement is true.

Class 7 Lines and Angles

ID : in-7-lines-and-angles [1] Class 7 Lines and Angles For more such worksheets visit www.edugain.com Answer t he quest ions (1) If AD and BD are bisectors of CAB and CBA respectively, f ind sum of angle