## NCERT Solutions for Chapter 5 Lines and Angles Class 7 Mathematics

Page No: 101**Exercise 5.1**

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

**Answer**

Complementary angle = 90Â° - given angle

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

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

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

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

**Answer**

Supplementary angle = 180Â° - given angle

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

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

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

3. Identify which of the following pairs of angles are complementary and which are supplementary: (i) 65Â°, 115Â°

(ii) 63Â°, 27Â°

(iii) 112Â°, 68Â°

(iv) 130âˆ˜, 50Â°

(v) 45Â°, 45Â°

(vi) 80Â°, 10Â°

**Answer**

If sum of two angles is 180Â°, then they are called supplementary angles. If sum of two angles is 90Â°, then they are called complementary angles.

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

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

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

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

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

(vi) 80Â°+10Â° =90Â°

These are complementary angles.

4. Find the angle which is equal to its complement:

**Answer**

Let one of the two equal complementary angles be x.

âˆ´ x + x = 90Â°

â‡’ 2x = 90Â°

â‡’x= 90Â°/2 = 45Â°

Thus, 45Â° is equal to its complement.

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

Answer

Let x be two equal angles of its supplement.

Therefore, x + x = 180Â° [Supplementary angles]

â‡’ 2x = 180 â‡’ x = 180Â°/2 = 90Â°

Thus, 90Â° is equal to its supplement.

6. In the given figure, âˆ 1 and âˆ 2 are supplementary angles. If âˆ 1 is decreased, what changes should take place in âˆ 2 so that both the angles still remain supplementary?

**Answer**

If âˆ 1 is decreased then, âˆ 2 will increase with the same measure, so that both the angles still remain supplementary.

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

(i) acute

(ii) obtuse

(iii) right?

**Answer**

(i) No, because sum of two acute angles is less than 180Â°

(ii) No, because sum of two obtuse angles is more than 180Â°

(iii) Yes, because sum of two right angles is 180Â°

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

**Answer**

Let the complementary angles be x and y i.e., x + y = 90Â°

It is given that x >45Â°

Adding y both sides, x+ y > 45Â° + y

â‡’ 90Â° > 45Â°+ y

â‡’ 90Â° - 45Â° > y

â‡’ y < 45Â°

Thus, its complementary angle is less than 45Â°

9. In the adjoining figure:

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

**Answer**

(i) Yes, in âˆ 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 âˆ 5 is âˆ COB.

10. Indicate which pairs of angles are:

Vertically opposite angles? Linear pairs?

**Answer**

(i) Vertically opposite angles, âˆ 1, âˆ 4; âˆ 5, âˆ 2 + âˆ 3.

(ii) Linear pairs âˆ 1, âˆ 5; âˆ 5, âˆ 4.

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

**Answer**

âˆ 1 and âˆ 2 are not adjacent angles because their vertex is not common.

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

**Answer**

(i) x = 55Â° [Vertically opposite angles]

Now 55Â° + y = 180Â°[Linear pair]

â‡’ y = 180Â° - 55Â° = 125Â°

Also, y=z=125Â° [Vertically opposite angles]

Thus, x=55Â°,y=125Â° and z=125Â°.

(ii) 40Â°+x+25Â°=180Â° [Angles on straight line]

â‡’ 65Â° +x=180Â°

â‡’ x=180Â°âˆ’65Â°= 115Â°

Now, 40Â°+y=180Â° [Linear pair]

â‡’ y=180Â°âˆ’40Â°=140Â° ....(i)

Also, y+z=180Â° [Linear pair]

â‡’ 140Â°+z=180Â° [From eq. (i)]

â‡’ z=180Â°âˆ’140Â°=40Â°

Thus, x=115Â°,y=140Â° and z=40Â°

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 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 ____________.

**Answer**

(i) 90Â°

(ii) 180Â°

(iii) supplementary

(iv) linear pair

(v) equal

(vi) obtuse angles

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 dÂ° nÂ°t fÂ°rm a linear pair.

**Answer**

(i) Obtuse vertically opposite angles means greater than 90Â° and equal âˆ AOD = âˆ BOC.

(ii) Adjacent cÂ°mplementary angles means angles have common vertex, common arm, non-common arms are on either side of common arm and sum of angles is 90Â°.

(iii) Equal supplementary angles means sum of angles is 180Â° and supplement angles are equal.

(iv) Unequal supplementary angles means sum of angles is 180Â° and supplement angles are unequal. i.e., âˆ AOE, âˆ EOC; âˆ AOD, âˆ DOC and âˆ AOB, âˆ 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., âˆ AOB, âˆ AOE; âˆ AOE, âˆ EOD and âˆ EOD, âˆ COD

Page No. 108

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

1. If aâˆ¥b, then âˆ 1 = âˆ 5.

2. If âˆ 4 = âˆ 6, then aâˆ¥b.

3. If âˆ 4 + âˆ 5 + 180âˆ˜, then aâˆ¥b.

**Answer**

(i) Given, aâˆ¥baâˆ¥b then âˆ 1 = âˆ 5 [Corresponding angles]

If two parallel lines are cut by a transversal, each pair of corresponding angles are equal in measure.

(ii) Given, âˆ 4 = âˆ 6, then aâˆ¥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, âˆ 4 + âˆ 5 = 180âˆ˜, then aâˆ¥b [Co-interior]

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.

2. In the adjoining 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.

**Answer**

(i) The pairs of corresponding angles:

âˆ 1, âˆ 5; âˆ 2, âˆ 6; âˆ 4, âˆ 8 and âˆ 3, âˆ 7

(ii) The pairs of alternate interior angles are:

âˆ 3, âˆ 5 and âˆ 2, âˆ 8

(iii) The pair of interior angles on the same side of the transversal:

âˆ 3, âˆ 8 and âˆ 2, âˆ 5

(iv) The vertically opposite angles are:

âˆ 1, âˆ 3; âˆ 2, âˆ 4; âˆ 6, âˆ 8 and âˆ 5, âˆ 7

3. In the adjoining figure, pâˆ¥q. Find the unknown angles.

**Answer**

Given, pâˆ¥q and cut by a transversal line.

âˆµ 125 ÌŠ+e=180 ÌŠ [Linear pair]

âˆ´ e=180 ÌŠâˆ’125 ÌŠ=55 ÌŠ â€¦.(i)

Now e=f=55 ÌŠ [Vertically opposite angles]

Also a=f=55 ÌŠ [Alternate interior angles]

a+b=180 ÌŠ [Linear pair]

â‡’ 55 ÌŠ+b=180 ÌŠ [From eq. (i)]

â‡’ b=180 ÌŠâˆ’55 ÌŠ=125 ÌŠ

Now a=c=55 ÌŠ and b=d=125 ÌŠ [Vertically opposite angles]

Thus, a=55 ÌŠ,b=125 ÌŠ,c=55 ÌŠ,d=125 ÌŠ,e=55 ÌŠ and f=55 ÌŠ.

4. Find the values of x in each of the following figures if lâˆ¥m.

**Answer**

(i) Given, lâˆ¥m and t is transversal line.

âˆ´ Interior vertically opposite angle between lines ll and t=110 ÌŠ.

âˆ´ 110 ÌŠ+x=180 ÌŠ [Supplementary angles]

â‡’ x=180 ÌŠâˆ’110 ÌŠ=70 ÌŠ

(ii) Given, lâˆ¥m and t is transversal line.

x+2x=180 ÌŠ [Interior opposite angles]

â‡’ 3x=180 ÌŠ

â‡’ x=180 ÌŠ/3=60 ÌŠ

(iii) Given, lâˆ¥m and aâˆ¥b

x=100 ÌŠ [Corresponding angles]

5. In the given figure, the arms of two angles are parallel. If Î”ABC = 70 ÌŠ, then find:

(i) DGC

(ii) DEF

**Answer**

(i) Given, AB âˆ¥ DE and BC is a transversal line and âˆ ABC=70 ÌŠ

âˆµ âˆ ABC = âˆ DGC [Corresponding angles]

âˆ´ âˆ DGC = 70 ÌŠ â€¦.(i)

(ii) Given, BC âˆ¥ EF and DE is a transversal line and âˆ DGC=70 ÌŠ

âˆµ âˆ DGC = âˆ DEF [Corresponding angles]

âˆ´ âˆ DEF = 70 ÌŠ [From eq. (i)]

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

**Answer**

(i) 126 ÌŠ+44 ÌŠ=170 ÌŠ

ll is not parallel to mm because sum of interior opposite angles should be 180âˆ˜.

(ii) 75 ÌŠ+75 ÌŠ=150 ÌŠ

ll is not parallel to mm because sum of angles does not obey the property of parallel lines.

(iii) 57 ÌŠ+123 ÌŠ=180 ÌŠ

ll is parallel to mm due to supplementary angles property of parallel lines.

(iv) 98 ÌŠ+72 ÌŠ=170 ÌŠ

ll is not parallel to mm because sum of angles does not obey the property of parallel lines.

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