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NCERT Solutions for class 7 Maths chapter 5 – Lines and Angles

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Exercise 5.1

Question 1.
Find the complement of each of the following angles:

Solution:
Since, the sum of the measures of an angle and its complement is 90°, therefore,

  1. The complement of an angle of measure 20° is the angle of (90° – 20°), f.e., 70°.
  2. The complement of an angle of measure 63° is the angle of (90° – 63°), i.e., 27°.
  3. The complement of an angle of measure 57° is the angle of (90° – 57°), i.e., 33°.

Question 2.
Find the supplement of each of the following angles:

Solution:

  1. Supplement of the angle 105° = 180° – 105° = 75°
  2. Supplement of the angle 87° = 180° – 87° = 93°
  3. Supplement of the angle 154° = 180° – 154° = 26°

Question 3.
Identify which of the following pairs of angles are complementary and which are supplementary.

  1. 65°, 115°
  2. 63°, 27°
  3. 112°, 68°
  4. 130°, 50°
  5. 45°,45°
  6. 80°, 10°.

Solution:

  1. Since, 65°+ 115° = 180°
    So, this pair of angles are supplementary.
  2. Since, 63°+ 27° = 90°
    So, this pair of angles are complementary.
  3. Since, 112° + 68° = 180°
    So, this pair of angles are supplementary.
  4. Since, 130°+50° = 180°
    So, this pair of angles are supplementary.
  5. Since, 45°+ 45° = 90°
    So, this pair of angles are complementary.
  6. Since, 80°+ 10° = 90°
    So, this pair of angles are complementary.

Question 4.
Find the angle which is equal to its complement.
Solution:
Let the measure of the angle be x°. Then, the measure of its complement is given to be x°.
Since, the sum of the measures of an angle and its complement is 90°, therefore,
x° + x° = 90°
⇒ 2x° = 90°
⇒ x° = 45°
Thus, the required angle is 45°.

Question 5.
Find the angle which is equal to its supplement.
Solution:
Let the measure of the angle be x°. Then,
a measure of its supplement = x°
Since, the sum of the measures of an angle and its supplement is 180°, therefore,
x° + x° = 180°
⇒ 2x° =180°
⇒ x° = 90°
Hence, the required angle is 90°.

Question 6.
In the given figure, ∠ 1 and ∠ 2 are supplementary angles.

If ∠1 is decreased, what changes should take place in ∠ 2 so that both the angles still remain supplementary?
Solution:
∠ 2 will increase as much as ∠ 1 decreases.

Question 7.
Can two angles be supplementary if both of them are:

  1. acute?
  2. obtuse?
  3. right?

Solution:

  1. No! two acute angles cannot be a supplement.
  2. No! Two obtuse angles cannot be supplementary.
  3. Yes! Two right angles are always supplementary.

Question 8.
An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45°.
Solution:
Since the sum of the measure of ah angle and its complement is 90°.
∴ The complement of an angle of measures 45° + x°,
where x > 0 is the angle of [90° – (45° + x°)] = 90° – 45° – x°= 45° – x°.
Clearly, 45° + x° > 45° – x°
Hence, the complement of an angle > 45° is less than 45°.

Question 9.
In the adjoining figure:

  1. Is ∠1 adjacent to ∠2 ?
  2. Is ∠ AOC adjacent to ∠ AOE?
  3. Do ∠ COE and ∠ EOD form a linear pair?
  4. Are ∠ BOD and ∠ DOA supplementary?
  5. Is ∠ 1 vertically opposite to ∠ 4?
  6. What is the vertically opposite angle of ∠ 5?

Solution:

  1. Yes ! ∠ 1 is adjacent to ∠ 2.
  2. No ! ∠ AOC is not adjacent to ∠ AOE.
  3. Yes! ∠ COE and ∠ EOD form a linear pair.
  4. Yes ! ∠ BOD and ∠ DOA are supplementary.
  5. Yes ! ∠ 1 is vertically opposite to ∠ 4.
  6. The vertically opposite angle of ∠ 5 is ∠ 2 + ∠ 3, i.e., ∠ COB.

Question 10.
Indicate which pairs of angles are:

  1. Vertically opposite angles.
  2. Linear pairs.

Solution:

  1. The pair of vertically opposite angles are ∠1, ∠4; ∠5, ∠2 + ∠3.
  2. The pair of linear angles are ∠1, ∠5; ∠4, ∠5.

Question 11.
In the following figure, is ∠ 1 adjacent to ∠ 2? Give reasons.

Solution:
∠1 is not adjacent to ∠2 because they have no common vertex.

Question 12.
Find the values of the angles x, y, and z in each of the following:

Solution:

Question 13.
Fill in the blanks:

  1. If two angles are complementary, then the sum of their measures is
  2. If two angles are supplementary, then the sum of their measures is
  3. Two angles forming a linear pair are
  4. If two adjacent angles are supplementary, they form a
  5. If two lines intersect at a point, then the vertically opposite angles are always
  6. If two lines intersect at a point, and if one pair of vertically opposite angles are acute angles, then the other pair of vertically opposite angles are

Solution:

  1. 90°
  2. 180°
  3. supplementary
  4. linear pair
  5. equal
  6. obtuse angles

Question 14.
In the adjoining figure, name the following pairs of angles.

  1. Obtuse vertically opposite angles
  2. Adjacent complementary angles
  3. Equal supplementary angles
  4. Unequal supplementary angles
  5. Adjacent angles that do not form a linear pair.

Solution:

  1. Obtuse vertically opposite angles are ∠AOD and ∠BOC.
  2. Adjacent complementary angles are ∠BOA and ∠AOE.
  3. Equal supplementary angles are ∠BOE and ∠EOD.
  4. Unequal supplementary angles are ∠BOA and ∠AOD, ∠BOC and ∠COD, ∠EOA, and ∠EOC.
  5. Adjacent angles that do not form a linear pair are ∠AOB and ∠AOE, ∠AOE and ∠EOD; ∠EOD and ∠COD.

Exercise 5.2

Question 1.
State the property that is used in each of the following statements?

  1. If a || b, then ∠ 1 = ∠ 5.
  2. If ∠ 4 = ∠ 6, then a || b.
  3. If ∠ 4 + ∠ 5 = 180°, then a || b.

Solution:

  1. Corresponding angle property.
  2. Alternate interior angle property.
  3. Interior angles on the same side of the transversal are supplementary.

Question 2.
In the following figure, identify:

  1. the pairs of corresponding angles.
  2. the pairs of alternate interior angles.
  3. the pairs of interior angles on the same side of the transversal.
  4. the vertically opposite angles.

Solution:

  1. ∠1, ∠5; ∠2, ∠6; ∠3, ∠7 and ∠4, ∠8 are four pairs of corresponding angles.
  2. ∠2, ∠8, and ∠3, ∠5 are two pairs of alternate interior angles.
  3. ∠2, ∠5, and ∠3, ∠8 are two pairs of interior angles on the same side of the transversal.
  4. ∠1, ∠3; ∠2, ∠4; ∠5, ∠7 and ∠6, ∠8 are four pairs of vertically opposite angles.

Question 3.
In the adjoining figure, p || q. Find the unknown angles.
Solution:
a = 55°, b = 125°, c = 55°, d = 125°, e = 55°, f = 55°.

Question 4.
Find the value of x in each of the following figures if l || m.

Solution:
(i) Since, l || m and t is a transversal.
∴ ∠x = (180° – 110°) = 70° [Corresponding angles, Linear pair]

(ii) if l || m and a is a transversal.
Then, ∠x = 1000 [Corresponding angles]

Question 5.
In the given figure, the arms of two angles are parallel. If ∠ ABC = 70°, then find

Solution:

  1. 70°
  2. 70°

Question 6.
In the given figures below, decide whether l is parallel to m.


Solution:

  1. l is not parallel to m
  2. l is not parallel to m
  3. l || m
  4. l is not parallel to m