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Chapter 14: Lines and Angles Exercise – 14.1 Question: 1 Write down each pair of adjacent angles shown in Figure Solution: The angles that have common vertex and a common arm are known as adjacent angles The adjacent angles are: ∠DOC and ∠BOC ∠COB and ∠BOA Question: 2 In figure, name all the pairs of adjacent angles. Solution: In fig (i), the adjacent angles are ∠EBA and ∠ABC ∠ACB and ∠BCF ∠BAC and ∠CAD In fig (ii), the adjacent angles are ∠BAD and ∠DAC ∠BDA and ∠CDA Question: 3 In fig , write down (i) each linear pair (ii) each pair of vertically opposite angles. Solution: (i) The two adjacent angles are said to form a linear pair of angles if their non – common arms are two opposite rays. ∠1 and ∠3 ∠1 and ∠2 ∠4 and ∠3 ∠4 and ∠2 ∠5 and ∠6 ∠5 and ∠7 ∠6 and ∠8 ∠7 and ∠8 (ii) The two angles formed by two intersecting lines and have no common arms are called vertically opposite angles. ∠1 and ∠4 ∠2 and ∠3 ∠5 and ∠8 ∠6 and ∠7 Question: 4 Are the angles 1 and 2 in figure adjacent angles? Solution: No, because they do not have common vertex. Question: 5 Find the complement of each of the following angles: (i) 35∘ (ii) 72∘ (iii) 45∘ (iv) 85∘ Solution: The two angles are said to be complementary angles if the sum of those angles is 90° Complementary angles for the following angles are: (i) 90° – 35° = 55° (ii) 90° – 72° = 18° (iii) 90° – 45° = 45° (iv) 90° – 85° = 5° Question: 6 Find the supplement of each of the following angles: (i) 70° (ii) 120° (iii) 135° (iv) 90° Solution: The two angles are said to be supplementary angles if the sum of those angles is 180° (i) 180° – 70° = 110° (ii) 180° – 120° = 60° (iii) 180° – 135° = 45° (iv) 180° – 90° = 90° Question: 7 Identify the complementary and supplementary pairs of angles from the following pairs (i) 25°, 65° (ii) 120°, 60° (iii) 63°, 27° (iv) 100°, 80° Solution: (i) 25° + 65° = 90° so, this is a complementary pair of angle. (ii) 120° + 60° = 180° so, this is a supplementary pair of angle. (iii) 63° + 27° = 90° so, this is a complementary pair of angle. (iv) 100° + 80° = 180° so, this is a supplementary pair of angle. Here, (i) and (iii) are complementary pair of angles and (ii) and (iv) are supplementary pair of angles. Question: 8 Can two obtuse angles be supplementary, if both of them be (i) obtuse? (ii) right? (iii) acute? Solution: (i) No, two obtuse angles cannot be supplementary Because, the sum of two angles is greater than 90 degrees so their sum will be greater than 180degrees. (ii) Yes, two right angles can be supplementary Because, 90° + 90° = 180° (iii) No, two acute angle cannot be supplementary Because, the sum of two angles is less than 90 degrees so their sum will also be less tha 90 degrees. Question: 9 Name the four pairs of supplementary angles shown in Fig. Solution: The supplementary angles are ∠AOC and ∠COB ∠BOC and ∠DOB ∠BOD and ∠DOA ∠AOC and ∠DOA Question: 10 In Figure, A, B, C are collinear points and ∠DBA = ∠EBA. (i) Name two linear pairs. (ii) Name two pairs of supplementary angles. Solution: (i) Linear pairs ∠ABD and ∠DBC ∠ABE and ∠EBC Because every linear pair forms supplementary angles, these angles are ∠ABD and ∠DBC ∠ABE and ∠EBC Question: 11 If two supplementary angles have equal measure, what is the measure of each angle? Solution: Let p and q be the two supplementary angles that are equal ∠p = ∠q So, ∠p + ∠q = 180° => ∠p + ∠p = 180° => 2∠p = 180° => ∠p = 180°/2 => ∠p = 90° Therefore, ∠p = ∠q = 90° Question: 12 If the complement of an angle is 28°, then find the supplement of the angle. Solution: Here, let p be the complement of the given angle 28° Therefore, ∠p + 28° = 90° => ∠p = 90° – 28° = 62° So, the supplement of the angle = 180° – 62° = 118° Question: 13 In Figure, name each linear pair and each pair of vertically opposite angles. Solution: Two adjacent angles are said to be linear pair of angles, if their non-common arms are two opposite rays. ∠1 and ∠2 ∠2 and ∠3 ∠3 and ∠4 ∠1 and ∠4 ∠5 and ∠6 ∠6 and ∠7 ∠7 and ∠8 ∠8 and ∠5 ∠9 and ∠10 ∠10 and ∠11 ∠11 and ∠12 ∠12 and ∠9 The two angles are said to be vertically opposite angles if the two intersecting lines have no common arms. ∠1 and ∠3 ∠4 and ∠2 ∠5 and ∠7 ∠6 and ∠8 ∠9 and ∠11 ∠10 and ∠12 Question: 14 In Figure, OE is the bisector of ∠BOD. If ∠1 = 70°, Find the magnitude of ∠2, ∠3, ∠4 Solution: Given, ∠1 = 70° ∠3 = 2(∠1) = 2(70°) = 140° ∠3 = ∠4 As, OE is the angle bisector, ∠DOB = 2(∠1) = 2(70°) = 140° ∠DOB + ∠AOC + ∠COB +∠DOB = 360° => 140° + 140° + 2(∠COB) = 360° Since, ∠COB = ∠AOD => 2(∠COB) = 360° – 280° => 2(∠COB) = 80° => ∠COB = 80°/2 => ∠COB = 40° Therefore, ∠COB = ∠AOB = 40° The angles are, ∠1 = 70°, ∠2 = 40°, ∠3 = 140°, ∠4 = 40° Question: 15 One of the angles forming a linear pair is a right angle. What can you say about its other angle? Solution: One of the Angle of a linear pair is the right angle (90°) Therefore, the other angle is => 180° – 90° = 90° Question: 16 One of the angles forming a linear pair is an obtuse angle. What kind of angle is the other? Solution: One of the Angles of a linear pair is obtuse, then the other angle should be acute, only then their sum will be 180°. Question: 17 One of the angles forming a linear pair is an acute angle. What kind of angle is the other? Solution: One of the Angles of a linear pair is acute, then the other angle should be obtuse, only then their sum will be 180°. Question: 18 Can two acute angles form a linear pair? Solution: No, two acute angles cannot form a linear pair because their sum is always less than 180°. Question: 19 If the supplement of an angle is 65°, then find its complement. Solution: Let x be the required angle So, => x + 65° = 180° => x = 180° – 65° = 115° But the complement of the angle cannot be determined. Question: 20 Find the value of x in each of the following figures
Write down each pair of adjacent angles shown in Figure
The angles that have common vertex and a common arm are known as adjacent angles
The adjacent angles are:
∠DOC and ∠BOC
∠COB and ∠BOA
In figure, name all the pairs of adjacent angles.
In fig (i), the adjacent angles are
∠EBA and ∠ABC
∠ACB and ∠BCF
∠BAC and ∠CAD
In fig (ii), the adjacent angles are
∠BAD and ∠DAC
∠BDA and ∠CDA
In fig , write down
(i) each linear pair
(ii) each pair of vertically opposite angles.
(i) The two adjacent angles are said to form a linear pair of angles if their non – common arms are two opposite rays.
∠1 and ∠3
∠1 and ∠2
∠4 and ∠3
∠4 and ∠2
∠5 and ∠6
∠5 and ∠7
∠6 and ∠8
∠7 and ∠8
(ii) The two angles formed by two intersecting lines and have no common arms are called vertically opposite angles.
∠1 and ∠4
∠2 and ∠3
∠5 and ∠8
∠6 and ∠7
Are the angles 1 and 2 in figure adjacent angles?
No, because they do not have common vertex.
Find the complement of each of the following angles:
(i) 35∘
(ii) 72∘
(iii) 45∘
(iv) 85∘
The two angles are said to be complementary angles if the sum of those angles is 90°
Complementary angles for the following angles are:
(i) 90° – 35° = 55°
(ii) 90° – 72° = 18°
(iii) 90° – 45° = 45°
(iv) 90° – 85° = 5°
Find the supplement of each of the following angles:
(i) 70°
(ii) 120°
(iii) 135°
(iv) 90°
The two angles are said to be supplementary angles if the sum of those angles is 180°
(i) 180° – 70° = 110°
(ii) 180° – 120° = 60°
(iii) 180° – 135° = 45°
(iv) 180° – 90° = 90°
Identify the complementary and supplementary pairs of angles from the following pairs
(i) 25°, 65°
(ii) 120°, 60°
(iii) 63°, 27°
(iv) 100°, 80°
(i) 25° + 65° = 90° so, this is a complementary pair of angle.
(ii) 120° + 60° = 180° so, this is a supplementary pair of angle.
(iii) 63° + 27° = 90° so, this is a complementary pair of angle.
(iv) 100° + 80° = 180° so, this is a supplementary pair of angle.
Here, (i) and (iii) are complementary pair of angles and (ii) and (iv) are supplementary pair of angles.
Can two obtuse angles be supplementary, if both of them be
(i) obtuse?
(ii) right?
(iii) acute?
(i) No, two obtuse angles cannot be supplementary
Because, the sum of two angles is greater than 90 degrees so their sum will be greater than 180degrees.
(ii) Yes, two right angles can be supplementary
Because, 90° + 90° = 180°
(iii) No, two acute angle cannot be supplementary
Because, the sum of two angles is less than 90 degrees so their sum will also be less tha 90 degrees.
Name the four pairs of supplementary angles shown in Fig.
The supplementary angles are
∠AOC and ∠COB
∠BOC and ∠DOB
∠BOD and ∠DOA
∠AOC and ∠DOA
In Figure, A, B, C are collinear points and ∠DBA = ∠EBA.
(i) Name two linear pairs.
(ii) Name two pairs of supplementary angles.
(i) Linear pairs
∠ABD and ∠DBC
∠ABE and ∠EBC
Because every linear pair forms supplementary angles, these angles are
If two supplementary angles have equal measure, what is the measure of each angle?
Let p and q be the two supplementary angles that are equal
∠p = ∠q
So,
∠p + ∠q = 180°
=> ∠p + ∠p = 180°
=> 2∠p = 180°
=> ∠p = 180°/2
=> ∠p = 90°
Therefore, ∠p = ∠q = 90°
If the complement of an angle is 28°, then find the supplement of the angle.
Here, let p be the complement of the given angle 28°
Therefore, ∠p + 28° = 90°
=> ∠p = 90° – 28°
= 62°
So, the supplement of the angle = 180° – 62°
= 118°
In Figure, name each linear pair and each pair of vertically opposite angles.
Two adjacent angles are said to be linear pair of angles, if their non-common arms are two opposite rays.
∠3 and ∠4
∠8 and ∠5
∠9 and ∠10
∠10 and ∠11
∠11 and ∠12
∠12 and ∠9
The two angles are said to be vertically opposite angles if the two intersecting lines have no common arms.
∠9 and ∠11
∠10 and ∠12
In Figure, OE is the bisector of ∠BOD. If ∠1 = 70°, Find the magnitude of ∠2, ∠3, ∠4
Given,
∠1 = 70°
∠3 = 2(∠1)
= 2(70°)
= 140°
∠3 = ∠4
As, OE is the angle bisector,
∠DOB = 2(∠1)
∠DOB + ∠AOC + ∠COB +∠DOB = 360°
=> 140° + 140° + 2(∠COB) = 360°
Since, ∠COB = ∠AOD
=> 2(∠COB) = 360° – 280°
=> 2(∠COB) = 80°
=> ∠COB = 80°/2
=> ∠COB = 40°
Therefore, ∠COB = ∠AOB = 40°
The angles are,
∠1 = 70°,
∠2 = 40°,
∠3 = 140°,
∠4 = 40°
One of the angles forming a linear pair is a right angle. What can you say about its other angle?
One of the Angle of a linear pair is the right angle (90°)
Therefore, the other angle is
=> 180° – 90° = 90°
One of the angles forming a linear pair is an obtuse angle. What kind of angle is the other?
One of the Angles of a linear pair is obtuse, then the other angle should be acute, only then their sum will be 180°.
One of the angles forming a linear pair is an acute angle. What kind of angle is the other?
One of the Angles of a linear pair is acute, then the other angle should be obtuse, only then their sum will be 180°.
Can two acute angles form a linear pair?
No, two acute angles cannot form a linear pair because their sum is always less than 180°.
If the supplement of an angle is 65°, then find its complement.
Let x be the required angle
=> x + 65° = 180°
=> x = 180° – 65°
= 115°
But the complement of the angle cannot be determined.
Find the value of x in each of the following figures
(i) Since, ∠BOA + ∠BOC = 180°
Linear pair:
=> 60° + x° = 180°
=> x° = 180° – 60°
=> x° = 120°
(ii) Linear pair:
=> 3x° + 2x° = 180°
=> 5x° = 180°
=> x° = 180°/5
=> x° = 36°
(iii) Linear pair,
Since, 35° + x° + 60° = 180°
=> x° = 180° – 35° – 60°
=> x° = 180° – 95°
=> x° = 85°
(iv) Linear pair,
83° + 92° + 47° + 75° + x° = 360°
=> x° + 297° = 360°
=> x° = 360° – 297°
=> x° = 63°
(v) Linear pair,
3x° + 2x° + x° + 2x° = 360°
=> 8x° = 360°
=> x° = 360°/8
=> x∘ = 45°
(vi) Linear pair:
3x° = 105°
=> x° = 105°/3
=> x° = 45°
In Fig. 22, it being given that ∠1 = 65°, find all the other angles.
∠1 = ∠3 are the vertically opposite angles
Therefore, ∠3 = 65°
Here, ∠1 + ∠2 = 180° are the linear pair
Therefore, ∠2 = 180° – 65°
∠2 = ∠4 are the vertically opposite angles
Therefore, ∠2 = ∠4 = 115°
And ∠3 = 65°
In Fig. 23 OA and OB are the opposite rays:
(i) If x = 25°, what is the value of y?
(ii) If y = 35°, what is the value of x?
∠AOC + ∠BOC = 180° – Linear pair
=> 2y + 5 + 3x = 180°
=> 3x + 2y = 175°
(i) If x = 25°, then
=> 3(25°) + 2y = 175°
=> 75° + 2y = 175°
=> 2y = 175° – 75°
=> 2y = 100°
=> y = 100°/2
=> y = 50°
(ii) If y = 35°, then
3x + 2(35°) = 175°
=> 3x + 70° = 175°
=> 3x = 175° – 70°
=> 3x = 105°
=> x = 105°/3
=> x = 35°
In Figure, write all pairs of adjacent angles and all the linear pairs.
Pairs of adjacent angles are:
∠DOA and ∠DOC
∠BOC and ∠COD
∠AOD and ∠BOD
∠AOC and ∠BOC
Linear pairs:
In Figure, find ∠x. Further find ∠BOC, ∠COD, ∠AOD
(x + 10)° + x° + (x + 20) ° = 180°
=> 3x° + 30° = 180°
=> 3x° = 180° – 30°
=> 3x° = 150°
=> x° = 150°/3
=> x° = 50°
Here,
∠BOC = (x + 20)°
= (50 + 20)°
= 70°
∠COD = 50°
∠AOD = (x + 10)°
= (50 + 10)°
= 60°
How many pairs of adjacent angles are formed when two lines intersect in a point?
If the two lines intersect at a point, then four adjacent pairs are formed and those are linear.
How many pairs of adjacent angles, in all, can you name in Figure?
There are 10 adjacent pairs
∠EOD and ∠DOC
∠COD and ∠BOC
∠AOB and ∠BOD
∠BOC and ∠COE
∠COD and ∠COA
∠DOE and ∠DOB
∠EOD and ∠DOA
∠EOC and ∠AOC
∠AOB and ∠BOE
In Figure, determine the value of x.
∠COB + ∠AOB = 180°
=> 3x° + 3x° = 180°
=> 6x° = 180°
=> x° = 180°/6
=> x° = 30°
In Figure, AOC is a line, find x.
∠AOB + ∠BOC = 180°
Linear pair
=> 2x + 70° = 180°
=> 2x = 180° – 70°
=> 2x = 110°
=> x = 110°/2
=> x = 55°
In Figure, POS is a line, find x.
Angles of a straight line,
∠QOP + ∠QOR + ∠ROS = 108°
=> 60° + 4x + 40° = 180°
=> 100° + 4x = 180°
=> 4x = 180° – 100°
=> 4x = 80°
=> x = 80°/4
=> x = 20°
In Figure, lines l_{1} and l_{2} intersect at O, forming angles as shown in the figure. If x = 45°, find the values of y, z and u.
Given that,
∠x = 45°
∠x = ∠z = 45°
∠y = ∠u
∠x + ∠y + ∠z + ∠u = 360°
=> 45° + 45° + ∠y + ∠u = 360°
=> 90° + ∠y + ∠u = 360°
=> ∠y + ∠u = 360° – 90°
=> ∠y + ∠u = 270°
=> ∠y + ∠z = 270°
=> 2∠z = 270°
=> ∠z = 135°
Therefore, ∠y = ∠u = 135°
So, ∠x = 45°,
∠y = 135°,
∠z = 45°,
∠u = 135°
In Figure, three coplanar lines intersect at a point O, forming angles as shown in the figure. Find the values of x, y, z and u
∠x + ∠y + ∠z + ∠u + 50° + 90° = 360°
Linear pair,
∠x + 50° + 90° = 180°
=> ∠x + 140° = 180°
=> ∠x = 180° – 140°
=> ∠x = 40°
∠x = ∠u = 40° are vertically opposite angles
=> ∠z = 90° is a vertically opposite angle
=> ∠y = 50° is a vertically opposite angle
Therefore, ∠x = 40°,
∠y = 50°,
∠z = 90°,
∠u = 40°
In Figure, find the values of x, y and z
∠y = 25° vertically opposite angle
∠x = ∠y are vertically opposite angles
∠x + ∠y + ∠z + 25° = 360°
=> ∠x + ∠z + 25° + 25° = 360°
=> ∠x + ∠z + 50° = 360°
=> ∠x + ∠z = 360° – 50°
=> 2∠x = 310°
=> ∠x = 155°
And, ∠x = ∠z = 155°
Therefore, ∠x = 155°,
∠y = 25°,
∠z = 155°
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Chapter 14: Lines and Angles Exercise – 14.2...