Class 7 Maths - Lines and Angles NCERT Solutions

Chapter 5: Lines and Angles (NCERT Solutions)

Exercise 5.1

Q1. Find the complement of each of the following angles:
(i) 20°
(ii) 63°
(iii) 57°

Two angles are said to be complementary if the sum of their measures is 90°.
Complement of an angle = 90° - Given angle.

(i) Complement of 20° = 90° - 20° = 70°

(ii) Complement of 63° = 90° - 63° = 27°

(iii) Complement of 57° = 90° - 57° = 33°

Q2. Find the supplement of each of the following angles:
(i) 105°
(ii) 87°
(iii) 154°

Two angles are said to be supplementary if the sum of their measures is 180°.
Supplement of an angle = 180° - Given angle.

(i) Supplement of 105° = 180° - 105° = 75°

(ii) Supplement of 87° = 180° - 87° = 93°

(iii) Supplement of 154° = 180° - 154° = 26°

Q3. 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°

(i) 65° + 115° = 180°. These are supplementary angles.

(ii) 63° + 27° = 90°. These are complementary angles.

(iii) 112° + 68° = 180°. These are supplementary angles.

(iv) 130° + 50° = 180°. These are supplementary angles.

(v) 45° + 45° = 90°. These are complementary angles.

(vi) 80° + 10° = 90°. These are complementary angles.

Q4. Find the angle which is equal to its complement.

Let the angle be x.
Its complement will also be x.
Since the sum of complementary angles is 90°:
x + x = 90°
2x = 90°
x = 90° / 2 = 45°

Q5. Find the angle which is equal to its supplement.

Let the angle be x.
Its supplement will also be x.
Since the sum of supplementary angles is 180°:
x + x = 180°
2x = 180°
x = 180° / 2 = 90°

Q6. 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?

Since ∠1 + ∠2 = 180° (supplementary angles).
If ∠1 is decreased by an amount, ∠2 must be increased by the same amount so that their sum remains 180°.

Q7. Can two angles be supplementary if both of them are:
(i) acute?
(ii) obtuse?
(iii) right?

(i) No. An acute angle is less than 90°. The sum of two acute angles will always be strictly less than 180° (e.g., 89° + 89° = 178°).

(ii) No. An obtuse angle is greater than 90°. The sum of two obtuse angles will always be strictly greater than 180° (e.g., 91° + 91° = 182°).

(iii) Yes. A right angle is exactly 90°. The sum of two right angles is exactly 180° (90° + 90° = 180°).

Q8. An angle is greater than 45°. Is its complementary angle greater than 45° or equal to 45° or less than 45°?

Let the given angle be A, such that A > 45°.
Let its complementary angle be B. Therefore, A + B = 90°.
Since A is greater than 45°, B must be less than 45° so that their sum exactly equals 90°.

Q9. In the adjoining figure:
(i) Is ∠1 adjacent to ∠2?
(ii) Is ∠AOC adjacent to ∠AOE?
(iii) Do ∠COE and ∠EOD form a linear pair?
(iv) Are ∠BOD and ∠DOA supplementary?
(v) Is ∠1 vertically opposite to ∠4?
(vi) What is the vertically opposite angle of ∠5?

(i) Yes. Because they have a common vertex O, a common arm OC, and their non-common arms OA and OE are on opposite sides of the common arm.

(ii) No. Because they have a common vertex O and a common arm OA, but their non-common arms OC and OE are on the same side of the common arm OA. Therefore, they do not satisfy all conditions of adjacent angles.

(iii) Yes. ∠COE and ∠EOD are adjacent angles whose non-common arms OC and OD form opposite rays (a straight line CD). So, they form a linear pair.

(iv) Yes. ∠BOD and ∠DOA form a linear pair on the straight line AB, so their sum is 180°. Thus, they are supplementary.

(v) Yes. Because they are formed by the intersection of two straight lines AB and CD, and they are opposite to each other at the vertex O.

(vi) The vertically opposite angle of ∠5 (∠AOD) is ∠COB (which is composed of ∠2 + ∠3).

Q10. Indicate which pairs of angles are:
(i) Vertically opposite angles.
(ii) Linear pairs.

(i) Vertically opposite angles:
- ∠1 and ∠4
- ∠5 and (∠2 + ∠3)

(ii) Linear pairs:
- ∠1 and ∠5 (on line containing ray opposite to 1)
- ∠4 and ∠5

Q11. In the following figure, is ∠1 adjacent to ∠2? Give reasons.

No. ∠1 is not adjacent to ∠2 because their vertices are different. Adjacent angles must share a common vertex.

Q12. Find the values of the angles x, y, and z in each of the following:
(i) Lines intersect at a point forming angles x, y, z and 55°.
(ii) Lines intersect forming angles 40°, x, 25°, y, z.

(i)
x and 55° are vertically opposite angles. So, x = 55°.
y and 55° form a linear pair. So, y + 55° = 180° ⇒ y = 180° - 55° = 125°.
z and y are vertically opposite angles. So, z = y = 125°.

(ii)
z and 40° are vertically opposite angles. So, z = 40°.
40° + x + 25° = 180° (Angles on a straight line) ⇒ x + 65° = 180° ⇒ x = 180° - 65° = 115°.
y and (x + 25°) are vertically opposite angles, or y and 40° form a linear pair (if arranged accordingly based on figure). Let's use linear pair: y + z = 180° ⇒ y + 40° = 180° ⇒ y = 140°.

Q13. Fill in the blanks:
(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 at 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 ______.

(i) 90°

(ii) 180°

(iii) supplementary

(iv) linear pair

(v) equal

(vi) obtuse angles

Exercise 5.2

Q1. 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.

(i) Property used: Corresponding angles property. If two parallel lines are cut by a transversal, then each pair of corresponding angles are equal in measure.

(ii) Property used: Alternate interior angles property (converse). If two lines are cut by a transversal such that a pair of alternate interior angles are equal, then the lines are parallel.

(iii) Property used: Consecutive interior angles property (converse). If two lines are cut by a transversal such that a pair of interior angles on the same side of the transversal are supplementary (sum to 180°), then the lines are parallel.

Q2. 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.

(i) Pairs of corresponding angles:
∠1 and ∠5 ; ∠2 and ∠6 ; ∠4 and ∠8 ; ∠3 and ∠7

(ii) Pairs of alternate interior angles:
∠3 and ∠5 ; ∠2 and ∠8

(iii) Pairs of interior angles on the same side of transversal:
∠3 and ∠8 ; ∠2 and ∠5

(iv) Vertically opposite angles:
∠1 and ∠3 ; ∠2 and ∠4 ; ∠5 and ∠7 ; ∠6 and ∠8

Q3. In the adjoining figure, p || q. Find the unknown angles.

Given line p || q and the intersecting line is a transversal.
125° + e = 180° (Linear pair)
e = 180° - 125° = 55°

f = e = 55° (Vertically opposite angles)

d = 125° (Corresponding angles to the given 125° angle)

a = f = 55° (Alternate interior angle to e, or corresponding to e)

c = a = 55° (Vertically opposite angles)

b = d = 125° (Vertically opposite angles)

Values: a = 55°, b = 125°, c = 55°, d = 125°, e = 55°, f = 55°

Q4. Find the value of x in each of the following figures if l || m.
(i) Angle adjacent to 110° (interior) is y.
(ii) Corresponding angle to x is 100°.

(i) Let the angle opposite to 110° be z (vertically opposite), so z = 110°.
Now z and x are interior angles on the same side of the transversal.
Since l || m, z + x = 180°
110° + x = 180°
x = 180° - 110° = 70°

(ii) Here, line l || line m, and line a is a transversal.
x and 100° are corresponding angles.
Therefore, x = 100°

Q5. In the given figure, the arms of two angles are parallel. If ∠ABC = 70°, then find:
(i) ∠DGC
(ii) ∠DEF

Given: AB || DE and BC || EF.
(i) Consider AB || DE and BC as transversal.
∠DGC and ∠ABC are corresponding angles.
So, ∠DGC = ∠ABC = 70°

(ii) Consider BC || EF and DE as transversal.
∠DEF and ∠DGC are corresponding angles.
So, ∠DEF = ∠DGC = 70°

Q6. In the given figures below, decide whether l is parallel to m.

(i) The sum of interior angles on the same side of transversal = 126° + 44° = 170°.
Since the sum is not 180°, line l is not parallel to line m.

(ii) Let the angle vertically opposite to 75° be x. So, x = 75°.
Sum of interior angles on the same side = x + 75° = 75° + 75° = 150°.
Since the sum is not 180°, line l is not parallel to line m.

(iii) Let the angle vertically opposite to 57° be y. y = 57°.
Sum of interior angles on the same side = 123° + y = 123° + 57° = 180°.
Since the sum is exactly 180°, line l is parallel to line m.

(iv) Let the angle vertically opposite to 72° be z. z = 72°.
Sum of interior angles on the same side = 98° + z = 98° + 72° = 170°.
Since the sum is not 180°, line l is not parallel to line m.

Class 7 Maths - Lines and Angles Practice Questions

Chapter 5: Lines and Angles (Practice Questions)

RD Sharma / Extra Practice

Q1. Two supplementary angles are in the ratio 4 : 5. Find the angles.

Let the angles be 4x and 5x.
Since they are supplementary:
4x + 5x = 180°
9x = 180°
x = 20°
Angles are 4(20°) = 80° and 5(20°) = 100°.

Q2. An angle is 20° more than its complement. Find the angle.

Let the complement be x.
Then the angle is x + 20°.
x + (x + 20°) = 90°
2x + 20° = 90°
2x = 70° ⇒ x = 35°.
The required angle = 35° + 20° = 55°.

Q3. Two complementary angles are in the ratio 2 : 7. Find the angles.

Let angles be 2x and 7x.
2x + 7x = 90°
9x = 90° ⇒ x = 10°.
Angles are 2(10°) = 20° and 7(10°) = 70°.

Q4. The measure of an angle is 30° less than its supplement. Find the angle.

Let the supplement be x.
Then the angle is x - 30°.
x + (x - 30°) = 180°
2x - 30° = 180°
2x = 210° ⇒ x = 105°.
The angle is 105° - 30° = 75°.

Q5. Among two supplementary angles, the measure of the larger angle is 44° more than the measure of the smaller. Find their measures.

Let the smaller angle be x.
The larger angle = x + 44°.
x + (x + 44°) = 180°
2x + 44° = 180°
2x = 136° ⇒ x = 68°.
Smaller angle = 68°, Larger angle = 68° + 44° = 112°.

Q6. Find the angle which is equal to one-third of its complement.

Let the complement be x, then angle = x/3.
x + (x/3) = 90°
4x/3 = 90°
4x = 270° ⇒ x = 67.5°.
The required angle = 67.5° / 3 = 22.5°.

Q7. Find the angle which is half of its supplement.

Let the supplement be x, then angle = x/2.
x + x/2 = 180°
3x/2 = 180°
3x = 360° ⇒ x = 120°.
The required angle = 120° / 2 = 60°.

Q8. A ray stands on a line, forming a linear pair of angles. If one of the angles is 65°, what is the other angle?

Since they form a linear pair, their sum is 180°.
Let the other angle be y.
65° + y = 180°
y = 180° - 65° = 115°.

Q9. In a linear pair of angles, if one angle is acute, what type of angle is the other?

Let the acute angle measure be x (where x < 90°).
The other angle in the linear pair is (180° - x).
Since x < 90°, (180° - x) must be greater than 90°.
Therefore, the other angle must be an obtuse angle.

Q10. Two straight lines intersect each other forming vertically opposite angles. If one pair of vertically opposite angles measures 45° each, find the measure of the other pair.

When two lines intersect, the adjacent angles form a linear pair.
Angle from the other pair + 45° = 180°.
Other angle = 180° - 45° = 135°.
Since vertically opposite angles are equal, the other pair measures 135° each.

Q11. In the adjoining figure, lines AB and CD intersect at O. If ∠AOC + ∠BOD = 130°, find ∠AOD.

∠AOC and ∠BOD are vertically opposite angles, so ∠AOC = ∠BOD.
∠AOC + ∠AOC = 130° ⇒ 2∠AOC = 130° ⇒ ∠AOC = 65°.
Now, ∠AOC and ∠AOD form a linear pair on line CD.
∠AOC + ∠AOD = 180°
65° + ∠AOD = 180° ⇒ ∠AOD = 115°.

Q12. What is the sum of angles around a point?

The total measure of all angles formed around a single point is 360°.

Q13. Two parallel lines are intersected by a transversal. If the measure of one of the consecutive interior angles is 75°, what is the measure of the other?

Consecutive interior angles (on the same side of transversal) between parallel lines are supplementary.
Let the other angle be y.
75° + y = 180°
y = 180° - 75° = 105°.

Q14. An angle is four times its complement. Find the measure of the angle.

Let the complement be x. Then the angle is 4x.
x + 4x = 90°
5x = 90° ⇒ x = 18°.
The angle is 4(18°) = 72°.

Q15. Can two obtuse angles form a linear pair?

No. An obtuse angle measures strictly greater than 90°.
The sum of two obtuse angles will always be strictly greater than 180°.
A linear pair must sum to exactly 180°.

Q16. Line l is parallel to line m, and cut by transversal t. If one of the alternate interior angles is 55°, what is the measure of its corresponding angle?

First, if an alternate interior angle is 55°, then the angle it alternates with is also 55°.
Since alternate interior angles are equal, and vertically opposite angles to interior ones form the corresponding pairs, the corresponding angle will also be 55°.

Q17. In a figure, two parallel lines are cut by a transversal. The interior angles on the same side are (2x - 8)° and (3x - 7)°. Find x.

Interior angles on the same side of transversal sum to 180°.
(2x - 8) + (3x - 7) = 180
5x - 15 = 180
5x = 195
x = 39.

Q18. Find the measure of an angle if seven times its complement is 10° less than three times its supplement.

Let the angle be x.
Complement = (90 - x), Supplement = (180 - x).
7(90 - x) = 3(180 - x) - 10
630 - 7x = 540 - 3x - 10
630 - 7x = 530 - 3x
630 - 530 = -3x + 7x
100 = 4x ⇒ x = 25°.

Q19. At what angle are two adjacent angles said to be a linear pair?

Two adjacent angles are termed a linear pair if their non-common arms form a straight line, which implies their sum must exactly be 180°.

Q20. If an angle is formed by a transversal intersecting two parallel lines, how many angles of the same measure are created?

When a transversal cuts two parallel lines, 8 angles are formed.
Unless the transversal is perpendicular, it creates exactly four acute angles that are all equal, and exactly four obtuse angles that are all equal.

Class 7 Maths - Lines and Angles Summary

Chapter 5: Lines and Angles (Concepts & Summary)

1. Basic Definitions

Line Segment: A part of a line with two end points.
Ray: A part of a line with one end point (starting point).
Line: A straight path extending infinitely in both directions. It has no end points.
Angle: An angle is formed when two lines (or rays or line segments) meet at a common point called the vertex. The symbol for angle is ∠.

2. Pairs of Angles

Complementary Angles: Two angles are complementary if the sum of their measures is exactly 90°. Each angle is called the complement of the other.
Supplementary Angles: Two angles are supplementary if the sum of their measures is exactly 180°. Each angle is called the supplement of the other.
Adjacent Angles: Two angles are adjacent if they have:
- A common vertex.
- A common arm.
- Non-common arms lying on either side of the common arm.
Linear Pair: A linear pair is a pair of adjacent angles whose non-common arms are opposite rays (they form a straight line). Angles in a linear pair are supplementary (sum = 180°).
Vertically Opposite Angles: When two lines intersect, the angles that are opposite to each other at the vertex are called vertically opposite angles. They are always equal in measure.

3. Intersecting Lines & Transversals

Intersecting Lines: Two lines that meet at a single point.
Transversal: A line that intersects two or more lines at distinct points.

4. Angles made by a Transversal

When a transversal intersects two lines, 8 angles are formed. These can be categorized as:

Interior Angles: The 4 angles that lie inside the two lines.
Exterior Angles: The 4 angles that lie outside the two lines.
Corresponding Angles: Pairs of angles in the same relative position at each intersection where the straight line crosses the others (e.g., above the line and to the right of the transversal).
Alternate Interior Angles: Pairs of interior angles that are on opposite sides of the transversal.
Alternate Exterior Angles: Pairs of exterior angles that are on opposite sides of the transversal.
Consecutive Interior Angles: Pairs of interior angles that are on the same side of the transversal (also called co-interior angles).

5. Transversal on Parallel Lines

If two parallel lines are cut by a transversal, then:

Each pair of Corresponding Angles are equal in measure.
Each pair of Alternate Interior Angles are equal in measure.
Each pair of Alternate Exterior Angles are equal in measure.
Each pair of Interior Angles on the same side of transversal is supplementary (i.e., their sum is 180°).

Note: The converse is also true. If any of the above conditions apply when two lines are cut by a transversal, then the two lines must be parallel.

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