# Complementary Angles

## Problems & Solutions

**Complementary Agnles:** When the sum of two angles is equal to 90^{o}, they are called complementary angles. In this case, both angles are called complementary to each others. For example 30^{o} and 60^{o}, 40^{o} and 50^{o}, 20^{o} and 70^{o}, etc. are complementary angle since their sum is equal to 90^{o}.

## Think Discuss and Write

### (NCERT Book – Math Class – VII – Page Number – 95)

**Question 1:** Can two acute angles complement each other?

**Answer:** Yes

**Explanation:** Acute angles are less than 90^{o}. Complementary angles are those angles whose sum is equal to 90^{o}. Thus, complementary angles are always acute angles. Hence, two acute angles can be complementary.

**Question 2:**Can two obtuse angles complement each other?

**Answer:** No

**Explanation:** Two obtuse angles cannot complement each other. Angles greater than 90^{o} are called obtuse angles. Hence, their sum would always be more than ninety degree. Thus, two obtuse angles cannot complement each other.

**Question 3:** Can two right angles complement each other?

**Answer:** No

**Explanation:** Angle equal to 90^{o} is called right angle. Sum of two right angles is 180° which is double the sum of two complementary angles. Hence, it is not possible for two right angles to complement each other.

#### Try These:

**Question 1:** Which pairs of following angles are complementary?

**Solution:** Sum of angles in fig (i) = 70° + 20° = 90°

Hence, they are complementary.

Sum of angles in fig (ii) = 75° + 25° = 100°> 90°

Hence, they are not complementary.

Sum of angles in fig (iii)= 48° + 52° = 100°> 90°

Hence, they are not complementary.

Sum of angles in fig (iv) = 35° + 55° = 90°

Hence, they are complementary.

**Question 2:** What is the measure of the complement of each of the following angles?

(i) 45º (ii) 65º (iii) 41º (iv) 54º

**Solution:** We know that the sum of two complementary angles is equal to 90^{o}.

Hence;

- Complementary of `45º= 90º- 45º= 45º`
- Complementary of `65º= 90º- 65º= 25º`
- Complementary of `41º= 90º- 41º= 49º`
- Complementary of `54º= 90º- 54º= 36º`

**Question 3:** The difference in the measures of two complementary angles is 12^{o}. Find the measures of the angles.

**Solution:** Let one angle is ‘m’

Therefore, as per question second angle will be `m + 12°`

We know that the sum of two complementary angle = 90^{o}Hence, m + m + 12^{o} = 90^{o}

Or, `2m + 12° = 90°`

Or, `2m = 90° - 12° = 78°`

Or, `m = 78° ÷ 2 = 39°`

Second angle `= m + 12° = 39° + 12° = 51°`

**Quesiton 4:** Find if the following angles make a complementary pair.

**Solution:** 27° + 63° = 90°

So, these angles make a complementary pair.

**Question 5:** Find the complementary angles for following angles:

(a) 15°

**Answer:** Let us assume the required complementary angle is x

So, `15° + x = 90°`

Or, `x = 90° - 15° = 75°`

Hence, the required complementary angle = 75°

(b) 35°

**Answer:** Let us assume the required complementary angle is x

So, `35° + x = 90°`

Or, `x = 90° - 35° = 55°`

Hence, the required complementary angle = 55°

(c) 45°

**Answer:** Let us assume the required complementary angle is x

So, `45° + x = 90°`

Or, `x = 90° - 45° = 45°`

Hence, the required complementary angle = 45°

(d) 60°

**Answer:** Let us assume the required complementary angle is x

So, `60° + x = 90°`

Or, `x = 90° - 60° = 30°`

Hence, the required complementary angle = 30°

(e) 70°

**Answer:** Let us assume the required complementary angle is x

So, `70° + x = 90°`

Or, `x = 90° - 70° = 20°`

Hence, the required complementary angle = 20°