**Lines and Angles**

**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,

- The complement of an angle of measure 20° is the angle of (90°-20°), f.e., 70°.
- The complement of an angle of measure 63° is the angle of (90°-63°), i.e., 27°.
- 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:**

Since, the sum of the measures of an angle and its supplement is 180°, therefore,

- The supplement of an angle of measvjre 105° is the angle of (180°-105°), i.e., 75°.
- The supplement of an angle of measure 87° is the angle of (180°-87°), i.e., 93°.
- The supplement of an angle of measure 154° is the angle of (180°-154°), i.e., 26°.

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

- 65°, 115°
- 63°, 27°
- 112°, 68°
- 130°, 50°
- 45°, 45°
- 80°, 10°

**Solution:**

- Since, 65°+ 115° = 180°

So, this pair of angles is supplementary. - Since, 63°+ 27° = 90°

So, this pair of angles is complementary. - Since, 112° + 68° = 1800

So, this pair of angles is supplementary. - Since, 130°+50° = 180°

So, this pair of angles is supplementary. - Since, 45°+ 45° = 90°

So, this pair of angles is complementary. - Since, 80°+ 10° = 90°

So, this pair of angles is 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,

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 with the same measure as the decrease in ∠1.

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

- acute?
- obtuse?
- right?

**Solution:**

- No
- No
- Yes

**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**

- Is ∠1 adjacent to ∠2 ?
- Is ∠AOC adjacent to ∠AOE?
- Do ∠COE and ∠EOD form a linear pair?
- Are ∠BOD and ∠DOA supplementary?
- Is ∠1 vertically opposite to Z4?
- What is the vertically opposite angle of ∠5

**Solution:**

- Yes
- No
- Yes
- Yes
- Yes
- ∠2 + ∠3 = ∠COB

**Question 10.****Indicate which pairs of angles are :**

- Vertically opposite angles.
- Linear pairs.

**Solution:**

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

**Question 11.****In the adjoining 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 :****(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 __________

**Solution:****(i)** 90°**(ii)** 180°**(iii)** supplementary**(iv)** linear pair**(v)** equal**(vi)** obtuse angles

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

- Obtuse vertically opposite angles.
- Adjacent complementary angles.
- Equal supplementary angles.
- Unequal supplementary angles.
- Adjacent angles that do not form a linear pair.

**Solution:**

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