NCERT Solutions Class 7 Mathematics Chapter 5 Lines and angles

Exercise 5.1

Q.1) Find the complement of the following angles

Sol.1) The sum of values of complementary angles is 90°.
(i) 20°
Complement = 90° – 20° = 70°
(ii) 63°
Complement = 90° – 63° = 27°
(iii) 57°
Complement = 90° – 57° = 33°

Q.2) Find the supplement of the following angles :

Sol.2) The sum of values of supplementary angles is 180°
(1) 95°
Supplement = 180° – 105° = 75°
(ii) 87°
Supplement = 180° – 87° = 93°
(iii) 154°
Supplement = 180° – 154° = 26°

Q.3) Identify which of the following pairs of angles are complementary and which are supplementary
i) 65°, 115° ii) 63°, 27° iii) 112°, 68°
iv) 130°, 50° v) 45°, 45° vi) 80°, 10°
Sol.3) The sum of the measures of complementary angles is 90° and that of supplementary angles is 180°.
(i) 65°, 115°
Sum of the measures of these angles = 65° + 115º = 180°
∴ These angles are supplementary angles.
(ii) 63°, 27°
Sum of the measures of these angles = 63° + 27º = 90°
∴ These angles are complementary angles.
(iii) 112°, 68°
Sum of the measures of these angles = 112° + 68º = 180°
∴ These angles are supplementary angles.
(iv) 130°, 50°
Sum of the measures of these angles = 130° + 50º = 180°
∴ These angles are supplementary angles.
(v) 45°, 45°
Sum of the measures of these angles = 45° + 45° = 90°
∴ These angles are complementary angles.
(vi) 80°, 10°
Sum of the measures of these angles = 80° + 10° = 90°
∴ These angles are complementary angles

Q.4) Find the angle which is equal to its complement
Sol.4) Let the angle be 𝑥.
Complement of this angle is also 𝑥.
The sum of the measures of a complementary angle pair is 90°.
∴ 𝑥 + 𝑥 = 90°
2𝑥 = 90°
𝑥 = 90°/2 = 45°

Q.5) Find the angle which is equal to its supplement.
Sol.5) Let the angle be 𝑥.
Supplement of this angle is also 𝑥.
The sum of the measures of a supplementary angle pair is 180°.
∴ 𝑥 + 𝑥 = 180°
2𝑥 = 180°
𝑥 = 90°

Q.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 angles still remain supplementary

Sol.6) ∠1 and ∠2 are supplementary angles.
If ∠1 is decreased, then ∠2 should also be increased by the same value so that this angle pair remains supplementary.

Q.7) Can two angles be supplementary if both of them are
i) acute? ii) obtuse? iii) right?
Sol.7) (i) No. Acute angle is always lesser than 90°. It can be observed that two angles, even of 90°, cannot add up to 180°. Therefore, two acute angles cannot be in a supplementary angle pair.
(ii) No. Obtuse angle is always greater than 90°. It can be observed that two angles, even of 90°, will always add up to more than 180°. Therefore, two obtuse angles cannot be in a supplementary angle pair.
(iii) Yes. Right angles are of 90° and 90° + 90° = 180°
Therefore, two right angles form a supplementary angle pair together.

Q.8) An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45°?
Sol.8) Let A and B are two angles making a complementary angle pair and A is greater than 45°
𝐴 + 𝐵 = 90°
𝐵 = 90° − 𝐴
Therefore, B will be lesser than 45°.

Q.9) In the adjoining figure :
(i) Is ∠1 adjacent to ∠2?
(ii) Is ∠𝐴𝑂𝐶adjacent to ∠𝐴𝑂𝐸?
(iii) Do ∠𝐶𝑂𝐸 and ∠𝐸𝑂𝐷 form a linear pair?
(iv) Are ∠𝐵𝑂𝐷and ∠𝐷𝑂𝐴 supplementary?
(v) Is ∠1 vertically opposite to ∠4?
(vi) What is the vertically opposite angle of ∠5?

Sol.9) (i) Yes, in ∠ 𝐴𝑂𝐸, OC is common arm.
(ii) No, they have no non-common arms on opposite side of common arm.
(iii) Yes, they form linear pair.
(iv) Yes, they are supplementary.
(v) Yes, they are vertically opposite angles.
(vi) Vertically opposite angles of ∠5 is ∠ 𝐶𝑂𝐵

Q.10) Indicate which pairs of angles are :
i) Vertically opposite angles.
ii) Linear pairs.

Sol.10) (i) Vertically opposite angles, ∠1 and ∠4; ∠5 and ∠2 + ∠3.
(ii) Linear pairs ∠1 and ∠5; ∠5 and ∠4.

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

Sol.11) ∠1 and ∠2 are not adjacent angles because their vertex is not common

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

Sol.12) (i) 𝑥 = 55° [Vertically opposite angles]
Now 55° + 𝑦 = 180° [Linear pair]
⇒ 𝑦 = 180° − 55° = 125°
Also 𝑦 = 𝑧 = 125° [Vertically opposite angles]
Thus, 𝑥 = 55° , 𝑦 = 125°and 𝑧 = 125°.
(ii) 40° + 𝑥 + 25° = 180° [Angles on straight line]
⇒ 65° + 𝑥 = 180°
⇒ 𝑥 = 180° − 65° = 115°
Now 40° + 𝑦 = 180° [Linear pair]
⇒ 𝑦 = 180 − 40 = 140 ……….(i)
Also 𝑦 + 𝑧 = 180° [Linear pair]
⇒ 140 + 𝑧 = 180° [From equation (i)]
⇒ 𝑧 = 180° − 140° = 40°
Thus, 𝑥 = 115°, 𝑦 = 140° & 𝑧 = 40°

Q.13) (i) If two angles are complementary, then the sum of their measures is _______.
(ii) If two angles are supplementary, then the sum of their measures is ________.
(iii) Two angles forming a linear pair are _________.
(iv) If two adjacent angles are supplementary, they form a __________
(v) If two lines intersect a point, then the vertically opposite angles are always _______.
(vi) 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 _________.
Sol.13) (i) 90° (ii) 180° (iii) Supplementary
(iv) Linear pair (v) Equal (vi) Obtuse angles

Q.14) In the adjoining figure, name the following pairs of angle.
i) Obtuse vertically opposite angles
ii) Adjacent complementary angles
iii) Equal supplementary angles
iv) Unequal supplementary angles
v) Adjacent angles that do not form a linear pair.

Sol.14) (i) Obtuse vertically opposite angles means greater than 90° and equal ∠ 𝐴𝑂𝐷 = ∠ 𝐵𝑂𝐶.
(ii) Adjacent complementary angles means angles have common vertex, common arm, non-common arms are on either side of common arm and sum of angles is 90° .
(iii) Equal supplementary angles means sum of angles is 180 and supplement angles are equal.
(iv) Unequal supplementary angles means sum of angles is 180° and supplement angles are unequal. i.e., ∠ 𝐴𝑂𝐸, ∠ 𝐸𝑂𝐶; ∠ 𝐴𝑂𝐷,∠ 𝐷𝑂𝐶 and ∠ 𝐴𝑂𝐵,∠ 𝐵𝑂𝐶
(v) Adjacent angles that do not form a linear pair mean, angles have common ray but the angles in a linear pair are not supplementary. i.e.,
∠ 𝐴𝑂𝐵,∠ 𝐴𝑂𝐸; ∠ 𝐴𝑂𝐸,∠ 𝐸𝑂𝐷 𝑎𝑛𝑑 ∠ 𝐸𝑂𝐷,∠ 𝐶𝑂𝐷

Exercise 5.2

Q.1) State the property that is used in each of the following statements:
(i) If 𝑎||𝑏, then ∠ 1 = ∠ 5.
(ii) If ∠ 4 = ∠ 6, then 𝑎||𝑏.
(iii) If ∠ 4 + ∠ 5 + 180° ,then 𝑎||𝑏.

Sol.1) (i) Given, 𝑎||𝑏, then ∠ 1 = ∠ 5 [Corresponding angles]
If two parallel lines are cut by a transversal, each pair of corresponding angles are equal in measure.
(ii) Given, ∠ 4 = ∠ 6, then 𝑎||𝑏 [Alternate interior angles]
When a transversal cuts two lines such that pairs of alternate interior angles are equal, the lines have to be parallel.
(iii) Given, ∠ 4 + ∠ 5 = 180°, then 𝑎||𝑏 [Co-interior Angles]
When a transversal cuts two lines, such that pairs of interior angles on the same side of transversal are supplementary, the lines have to be parallel

Q.2) In the adjoining figure, identify
i) the pairs of corresponding angles.
ii) the pairs of alternate interior angles.
iii) the pairs of interior angles on the same side of the transversal.
iv) the vertically opposite angles.

Sol.2) (i) The pairs of corresponding angles: ∠ 1,∠ 5; ∠ 2,∠ 6; ∠ 4, ∠ 8 and ∠ 3,∠ 7
(ii) The pairs of alternate interior angles are: ∠ 3,∠ 5 and ∠ 2,∠ 8
(iii) The pair of interior angles on the same side of the transversal: ∠ 3, ∠ 8 and ∠ 2,∠ 5
(iv) The vertically opposite angles are: ∠ 1,∠ 3; ∠ 2,∠ 4; ∠ 6,∠ 8 and ∠ 5,∠ 7

Q.3) In the adjoining figure, 𝑝||𝑞. Find the unknown angles.

Sol.3) Given, 𝑝||𝑞 and cut by a transversal line.
125° + 𝑒 = 180° [Linear pair]
∴ 𝑒 = 180° − 125° = 55° ……….(i)
Now 𝑒 = 𝑓 = 55° [Vertically opposite angles]
Also 𝑎 = 𝑓 = 55° [Alternate interior angles]
𝑎 + 𝑏 = 180° [Linear pair]
⇒ 55° + 𝑏 = 180° [From equation (i)]
⇒ 𝑏 = 180° − 55° = 125°
Now 𝑎 = 𝑐 = 55°and 𝑏 = 𝑑 = 125° [Vertically opposite angles]
Thus, 𝑎 = 55°, 𝑏 = 125°, 𝑐 = 55°, 𝑑 = 125°, 𝑒 = 55°and 𝑓 = 55°

Q.4) Find the values of 𝑥 in each of the following figures if 𝑙||𝑚.

Sol.4) (i) Given, l||m and t is transversal line.
∴ Interior vertically opposite angle between lines 𝑙 and 𝑡 = 110
∴ 110° + 𝑥 = 180° [Supplementary angles]
⇒ 𝑥 = 180° − 110° = 70°
(ii) Given, 𝑙||𝑚 and t is transversal line.
𝑥 + 2𝑥 = 180° [Interior opposite angles]
⇒ 3𝑥 = 180°
⇒ 𝑥 = 180°/3 = 60°
(iii) Given, 𝑙||𝑚 and 𝑎||𝑏. 𝑥 = 100°[Corresponding angles]

Q.5) In the given figure, the arms of two angles are parallel. If Δ𝐴𝐵𝐶 = 70°, then find:
(i) ∠ 𝐷𝐺𝐶    (ii) ∠ 𝐷𝐸𝐹

Sol.5) (i) Given, 𝐴𝐵 || 𝐷𝐸 and 𝐵𝐶 is a transversal line and ∠ 𝐴𝐵𝐶 = 70°,
∵ ∠ 𝐴𝐵𝐶 = ∠ 𝐷𝐺𝐶 [Corresponding angles]
∴ ∠ 𝐷𝐺𝐶 = 70°, ……….(i)
(ii) Given, 𝐵𝐶 || 𝐸𝐹 and 𝐷𝐸 is a transversal line and ∠ 𝐷𝐺𝐶 = 70°,
∠ 𝐷𝐺𝐶 = ∠ 𝐷𝐸𝐹 [Corresponding angles]
∴ ∠ 𝐷𝐸𝐹 = 70°, [From equation (i)]

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

Sol.6) (i) 126° + 44° = 170°
𝑙 || 𝑚 because sum of interior opposite angles should be 180 .
(ii) 75° + 75° = 150°
𝑙 || 𝑚 because sum of angles does not obey the property of parallel lines.
(iii) 57° + 13° = 180°
𝑙 || 𝑚 due to supplementary angles property of parallel lines.
(iv) 98° + 72° = 170°
𝑙 is not parallel to m because sum of angles does not obey the property of parallel lines.