ID : ae-9-Lines-and-Angles [1]

Grade 9 Lines and Angles For more such worksheets visit www.edugain.com

Answer t he quest ions (1)

What is the value of the supplement of the complement of 61°?

(2)

If AB and PQ are parallel, compute the angle Z .

(3)

If AB and CD are parallel, f ind the value of X

(4) If two horizontal lines are parallel, f ind the value of angle x.

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [2]

Choose correct answer(s) f rom given choice (5)

If OD is perpendicular to AB, and ∠DOC = 25°, f ind (∠BOC - ∠AOC).

a. 55°

b. 40°

c. 45°

d. 50°

(6) If AP and BP are bisectors of angles ∠CAB and ∠CBD respectively, f ind the angle ∠APB.

a. 25°

b. 40°

c. 35°

d. 30°

(7) If angles of a triangle are in ratio 5:2:8, the triangle is

(8)

a. an acute angled triangle

b. an obtuse angled triangle

c. a right triangle

d. an isosceles triangle

If AB and CD are parallel, f ind the value of angle x.

a. 120°

b. 70°

c. 110°

d. 100°

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [3]

(9) What is the angle between hour and minute hands, when clock shows 8:00 o'clock ? a. 120°

b. 90°

c. 110°

d. 150°

(10) Lines AB and CD intersect at O. If ∠AOC + ∠BOE = 130° and ∠BOD = 70°, f ind ∠BOE.

a. 60°

b. 75°

c. 50°

d. 70°

(11) If AB and CD are parallel, f ind the value of x+y.

a. 55°

b. 95°

c. 75°

d. 85°

(12) If AB and DE are parallel, f ind the value of ∠ACB

a. 115°

b. 105°

c. 110°

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [4]

Fill in t he blanks (13)

Value of angle p is

°

(14)

Angle x =

°

Check True/False (15) A triangle can have two obtuse angles. T rue False

© 2016 Edugain (www.edugain.com). All Rights Reserved

(C) 2016 Edugain (www.Edugain.com)

Many more such worksheets can be generated at www.edugain.com

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [5]

Answers (1)

151° Step 1 If you look at the question caref ully, you will notice that f irst of all we have to f ind the complement of 61° , then f ind the supplement of the complement of 61°. Step 2 T he sum of the complementary angles is 90°. T heref ore the complement of 61° = 90° - 61° = 29° Step 3 T he sum of supplementary angles is 180°. T heref ore, the supplement of 29° = 180° - 29° = 151° Step 4 T heref ore the value of the supplement of the complement of 61° is 151° .

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [6]

(2)

269° Step 1

Construction: Draw a line MN which is parallel to the line PQ and the line AB. Angle Z = angle Z 1 + angle Z 2. Step 2 Angle P = 45°, angle Z 1 and angle P are alternate angles, angle Z 1 = angle P, ⇒ angle Z 1 = 45°. Step 3 Similarly, angle A = 46°, angle Z 2 and angle A are alternate angles, angle Z 2 = angle A, ⇒ angle Z 2 = 46°. Step 4 Angle Z = 360° - angle Z 1 - angle Z 2 = 360° - 45° - 46° = 360° - 91° = 269°. Step 5 T heref ore, the angle Z is 269°.

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [7]

(3)

25 ° Step 1 Parallel line AB and CD are intersected by a transversal as shown below,

Here angle ∠P and ∠Q are complementary angles. i.e. ∠P + ∠Q = 180° Step 2 On comparing given angles with ∠P and ∠Q, 2x + 3x + 55° = 180° ⇒ 5x = 125° ⇒ x = 25°

(4) 70°

When a straight line cuts any two parallel lines, its Alternate Angles are equal. Since angles x and 70° are Alternate angles. T heref ore, x = 70° ...[Alternate angles of two parallel lines are equal]

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [8]

(5)

d. 50° Step 1 According to question ∠DOC = 25° and OD is perpendicular to AB. T heref ore ∠AOD = 90° and ∠BOD = 90°. Step 2 ∠DOC + ∠AOC = ∠AOD ⇒ 25° + ∠AOC = 90° [Since ∠AOD = 90° and ∠DOC = 25°] ⇒ ∠AOC = 90° - 25° ⇒ ∠AOC = 65° Step 3 Now ∠BOC - ∠AOC = ∠BOD + ∠DOC - ∠AOC [Since ∠BOC = ∠BOD + ∠DOC] = 90° + 25° - 65° [Since ∠BOD = 90°, ∠DOC = 25° and ∠AOC = 65°] = 50° Step 4 T heref ore ∠BOC - ∠AOC = 50°

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [9]

(6) c. 35° Step 1

As per the question ∠CBD is exterior angle of the triangle ABC and we know that an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. T heref ore, ∠CBD = ∠CAB + 70° ----(1) Step 2 In triangle ABC, ∠CAB + ∠ABC + ∠ACB = 180° ⇒ ∠CAB + ∠ABC = 180° - 70° ⇒ ∠CAB + ∠ABC = 110° -----(2) Step 3 It is given that AP and BP are bisectors of angles ∠CAB and ∠CBD respectively. T heref ore, ∠PAB = ∠CAB/2 -----(3) ∠CBP = ∠CBD/2 -----(4) Step 4 Now, in triangle ABP, ∠PAB + ∠ABP + ∠APB = 180° ...[Since the sum of all three angles of a triangle is 180°] ⇒ ∠APB = 180° - ∠PAB - ∠ABP ⇒ ∠APB = 180° - ∠CAB/2 - ∠ABP ..[From equation (3), ∠PAB = ∠CAB/2] ⇒ ∠APB = 180° - ∠CAB/2 - (∠ABC + ∠CBP) ⇒ ∠APB = 180° - ∠CAB/2 - (∠ABC + ∠CBD/2) ...[From equation (4), ∠CBP = ∠CBD/2] ⇒ ∠APB = 180° - ∠CAB/2 - {∠ABC + (∠CAB + 70°)/2} ...[From equation (1)] ⇒ ∠APB = 180° - ∠CAB/2 - (∠ABC + ∠CAB/2 + 35°) ⇒ ∠APB = 180° - ∠CAB/2 - ∠ABC - ∠CAB/2 - 35° ⇒ ∠APB = 145° - (∠CAB + ∠ABC) ⇒ ∠APB = 145° - 110° ...[From equation (2)] ⇒ ∠APB = 35° Step 5 Hence, the value of angle ∠APB is 35°.

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [10]

(7) b. an obtuse angled triangle Step 1 According to the question, all angles of the triangle are in ratio 5:2:8. We can assume three angles of the triangle to be 5x, 2x and 8x where x is common f actor. Step 2 We know that the sum of the three angles of a triangle is 180°. T heref ore, 5x + 2x + 8x = 180° ⇒ 15x = 180° ⇒x=

180 15

Now, 5x = 5 ×

180

= 60,

15 2x = 2 ×

180

= 24 and

15 8x = 8 ×

180

= 96.

15 T heref ore, the three angles of the triangle are 60°, 24° and 96°. Step 3 Since, one of the angle of the triangle is greater than 90°, the triangle is an obtuse angled triangle.

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [11]

(8)

c. 110° Step 1 It is given that line AB and CD and parallel lines and the third line (say EF) cuts the lines AB and CD at certain angle as shown in the f igure above. Let us redraw the f igure as below:

∠a = ∠c (vertically opposite angles) ∠c = ∠e (alternate interior angles) T heref ore we can write, ∠a = ∠c = ∠e = ∠g Again, ∠b = ∠d (vertically opposite angles) ∠d = ∠f (alternate interior angles) T heref ore we can write, ∠b = ∠d = ∠f = ∠h We know that sum of two adjacent angle is equal to 180°. T heref ore, f rom the diagram, you can write, ∠a + ∠b = 180°, ∠b + ∠c = 180°, ∠c + ∠d = 180°, ∠d + ∠a = 180° Step 2 Here, ∠e = 70° and ∠d = x ∠e + ∠h = 180° ⇒ 70° + ∠h = 180° ⇒ ∠h = 180° - 70° ⇒ ∠h = 110° As ∠h is equal to ∠d, x is 110°. Step 3 T heref ore, the value of x is 110°. (C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [12]

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [13]

(10) a. 60° Step 1

If you look at the given f igure caref ully, you will notice that AB and CD are straight lines. ∠AOC + ∠BOE = 130° and ∠BOD = 70°. Step 2 T he angles of straight line add up to 180°. Line AB is a straight line, theref ore we can say that ∠AOC + ∠COE + ∠BOE = 180° ⇒ ∠AOC + ∠BOE + ∠COE = 180° ⇒ 130° + ∠COE = 180° [Since ∠AOC + ∠BOE = 130°] ⇒ ∠COE = 180° - 130° ⇒ ∠COE = 50° Step 3 CD is also a straight line, theref ore ∠COE + ∠BOE + ∠BOD = 180° ⇒ 50° + ∠BOE + 70° = 180° [Since ∠COE = 50° and ∠BOD = 70°] ⇒ ∠BOE + 120° = 180° ⇒ ∠BOE = 180° - 120° ⇒ ∠BOE = 60°. Step 4 Now ∠BOE = 60°.

(11) c. 75° Step 1 According to question AB and CD are parallel, theref ore ∠x = 45° [Alternate interior angles] ∠y = 30° [Alternate interior angles] Step 2 Now the value of x+y = 45° + 30° = 75°

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [14]

(12) c. 110° Step 1

If you look at the f igure caref ully, you will notice that the angle ∠BAC = 25°, ∠BDE = 45°. Step 2 According to question AB and DE are parallel. T heref ore the angles ∠ABC and ∠BDE are alternate interior angles. ∠ABC = ∠BDE [Alternate interior angles] ⇒ ∠ABC = 45° Step 3 T he sum of all three angles of a triangle is 180°. Now in triangle ABC, ∠ABC + ∠BAC + ∠ACB = 180° ⇒ 45° + 25° + ∠ACB = 180° [Since ∠ABC = 45° and ∠BAC = 25°] ⇒ 70° + ∠ACB = 180° ⇒ ∠ACB = 180° - 70° ⇒ ∠ACB = 110° Step 4 T heref ore the value of ∠ACB is 110°.

(13)

56 Step 1 If you look at the angles 106° and p + 50°, these are opposite angles Step 2 We know that opposite angles are equal. T heref ore p + 50° = 106° ⇒ p = 106° - 50° ⇒ p = 56°

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [15]

(14)

40 Step 1

If you look at the f igure caref ully, you will notice that line AB is a straight line. Step 2 T he angles of straight line add up to 180°. Line AB is a straight line, theref ore 80° + x + 60° = 180° ⇒ 140° + ∠x = 180° ⇒ ∠x = 180° - 140° ⇒ ∠x = 40°. Step 3 T heref ore the value of angle x is 40°.

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited

ID : ae-9-Lines-and-Angles [16]

(15) False Step 1

Let's consider the triangle ABC in the f igure above. Since we know that the sum of all three angles of a triangle is 180°, in ΔABC: ∠A + ∠B + ∠C = 180°. Step 2 Let's assume that ∠A of the ΔABC is an obtuse angle. T hat is, ∠A > 90°. Now, ∠A + ∠B + ∠C = 180° ⇒ ∠B + ∠C = 180° - ∠A ⇒ ∠B + ∠C < 90° (Since ∠A > 90°) Step 3 We just saw that the sum of ∠B and ∠C of the ΔABC is less than 90°. T heref ore, we can say that the ∠B and the ∠C must be acute angles and the statement "A triangle can have two obtuse angles" is False.

(C) 2016 Edugain (www.Edugain.com)

Personal use only, commercial use is strictly prohibited