Fundamental Concepts of Geometry — Exercise 14(D)

Write the complement angle of:

45°

Answer

The complement of an angle = 90° − the angle.

Complement of 45° = 90° − 45° = 45°

Hence, the complement angle of 45° is 45°.

Write the complement angle of:

x°

Answer

The complement of an angle = 90° − the angle.

Complement of x° = (90 − x)°

Hence, the complement angle of x° is (90 − x)°.

Write the complement angle of:

(x - 10)°

Answer

The complement of an angle = 90° − the angle.

Complement of (x − 10)° = 90° − (x − 10)° = 90° − x° + 10° = (100 − x)°

Hence, the complement angle of (x − 10)° is (100 − x)°.

Write the complement angle of:

20° + y°

Answer

The complement of an angle = 90° − the angle.

Complement of (20° + y°) = 90° − (20° + y°) = 90° − 20° − y° = (70 − y)°

Hence, the complement angle of (20° + y°) is (70 − y)°.

Write the supplement angle of:

49°

Answer

The supplement of an angle = 180° − the angle.

Supplement of 49° = 180° − 49° = 131°

Hence, the supplement angle of 49° is 131°.

Write the supplement angle of:

111°

Answer

The supplement of an angle = 180° − the angle.

Supplement of 111° = 180° − 111° = 69°

Hence, the supplement angle of 111° is 69°.

Write the supplement angle of:

(x - 30)°

Answer

The supplement of an angle = 180° − the angle.

Supplement of (x − 30)° = 180° − (x − 30)° = 180° − x° + 30° = (210 − x)°

Hence, the supplement angle of (x − 30)° is (210 − x)°.

Write the supplement angle of:

20° + y°

Answer

The supplement of an angle = 180° − the angle.

Supplement of (20° + y°) = 180° − (20° + y°) = 180° − 20° − y° = (160 − y)°

Hence, the supplement angle of (20° + y°) is (160 − y)°.

Find the complement angle of:

$\dfrac{1}{2}$ of 60°

Answer

As, $\dfrac{1}{2} \text{ of } 60\degree = \dfrac{1}{2} \times 60\degree = 30\degree$

The complement of an angle = 90° − the angle.

Complement of 30° = 90° − 30° = 60°

Hence, the complement of 30° is 60°.

Find the complement angle of:

$\dfrac{1}{5}$ of 160°

Answer

As, $\dfrac{1}{5} \text{ of } 160\degree = \dfrac{1}{5} \times 160\degree = 32\degree$

The complement of an angle = 90° − the angle.

Complement of 32° = 90° − 32° = 58°

Hence, the complement of 32° is 58°.

Find the complement angle of:

$\dfrac{2}{5}$ of 70°

Answer

As, $\dfrac{2}{5} \text{ of } 70\degree = \dfrac{2}{5} \times 70\degree = 28\degree$

The complement of an angle = 90° − the angle.

Complement of 28° = 90° − 28° = 62°

Hence, the complement of 28° is 62°.

Find the complement angle of:

$\dfrac{1}{6}$ of 90°

Answer

As, $\dfrac{1}{6} \text{ of } 90\degree = \dfrac{1}{6} \times 90\degree = 15\degree$

The complement of an angle = 90° − the angle.

Complement of 15° = 90° − 15° = 75°

Hence, the complement of 15° is 75°.

Find the supplement angle of:

50% of 120°

Answer

As, 50% of 120° = $\dfrac{50}{100} \times 120\degree = 60\degree$

The supplement of an angle = 180° − the angle.

Supplement of 60° = 180° − 60° = 120°

Hence, the supplement of 60° is 120°.

Find the supplement angle of:

$\dfrac{1}{3}$ of 150°

Answer

As, $\dfrac{1}{3} \text{ of } 150\degree = \dfrac{1}{3} \times 150\degree = 50\degree$

The supplement of an angle = 180° − the angle.

Supplement of 50° = 180° − 50° = 130°

Hence, the supplement of 50° is 130°.

Find the supplement angle of:

60% of 100°

Answer

As, 60% of 100° = $\dfrac{60}{100} \times 100\degree = 60\degree$

The supplement of an angle = 180° − the angle.

Supplement of 60° = 180° − 60° = 120°

Hence, the supplement of 60° is 120°.

Find the supplement angle of:

$\dfrac{3}{4}$ of 160°

Answer

As, $\dfrac{3}{4} \text{ of } 160\degree = \dfrac{3}{4} \times 160\degree = 120\degree$

The supplement of an angle = 180° − the angle.

Supplement of 120° = 180° − 120° = 60°

Hence, the supplement of 120° is 60°.

Find the angle that is equal to its complement?

Answer

Let the required angle be x.

As the angle is equal to its complement,

⇒ x = 90° − x

⇒ x + x = 90°

⇒ 2x = 90°

⇒ x = $\dfrac{90\degree}{2}$

⇒ x = 45°

Hence, the required angle is 45°.

Find the angle that is equal to its supplement?

Answer

Let the required angle be x.

As the angle is equal to its supplement,

⇒ x = 180° − x

⇒ x + x = 180°

⇒ 2x = 180°

⇒ x = $\dfrac{180\degree}{2}$

⇒ x = 90°

Hence, the required angle is 90°.

Two complementary angles are in the ratio 7 : 8. Find the angles.

Answer

As the angles are in the ratio 7 : 8.

Let the angles be 7x and 8x.

As the given angles are complementary, their sum is 90°.

⇒ 7x + 8x = 90°

⇒ 15x = 90°

⇒ x = $\dfrac{90\degree}{15}$

⇒ x = 6°

So, the angles are 7x = 7 × 6° = 42° and 8x = 8 × 6° = 48°.

Hence, the angles are 42° and 48°.

Two supplementary angles are in the ratio 7 : 11. Find the angles.

Answer

As the angles are in the ratio 7 : 11.

Let the angles be 7x and 11x.

As the given angles are supplementary, their sum is 180°.

⇒ 7x + 11x = 180°

⇒ 18x = 180°

⇒ x = $\dfrac{180\degree}{18}$

⇒ x = 10°

So, the angles are 7x = 7 × 10° = 70° and 11x = 11 × 10° = 110°.

Hence, the angles are 70° and 110°.

The measures of two complementary angles are (2x - 7)° and (x + 4)°. Find x.

Answer

As the given angles are complementary, their sum is 90°.

⇒ (2x − 7)° + (x + 4)° = 90°

⇒ 2x° - 7° + x° + 4° = 90°

⇒ 3x° − 3° = 90°

⇒ 3x° = 90° + 3°

⇒ 3x° = 93°

⇒ x° = $\dfrac{93\degree}{3}$

⇒ x° = 31°

⇒ x = 31

Hence, x = 31.

The measures of two supplementary angles are (3x + 15)° and (2x + 5)°. Find x.

Answer

As the given angles are supplementary, their sum is 180°.

⇒ (3x + 15)° + (2x + 5)° = 180°

⇒ 3x° + 15° + 2x° + 5° = 180°

⇒ 5x° + 20° = 180°

⇒ 5x° = 180° − 20°

⇒ 5x° = 160°

⇒ x° = $\dfrac{160\degree}{5}$

⇒ x° = 32°

⇒ x = 32

Hence, x = 32.

For an angle x°, find:

(i) the complementary angle

(ii) the supplementary angle.

(iii) the value of x° if its supplementary angle is three times its complementary angle.

Answer

(i) Two angles are said to be complementary if their sum is 90°, so the complement of an angle = 90° − the angle.

Complement of x° = 90° − x° = (90 − x)°

Hence, the complementary angle is (90 − x)°.

(ii) Two angles are said to be supplementary if their sum is 180°, so the supplement of an angle = 180° − the angle.

Supplement of x° = 180° − x° = (180 − x)°

Hence, the supplementary angle is (180 − x)°.

(iii) As the supplementary angle is three times the complementary angle,

⇒ (180 − x)° = 3(90 − x)°

⇒ 180° − x° = 270° − 3x°

⇒ 3x° − x° = 270° − 180°

⇒ 2x° = 90°

⇒ x° = $\dfrac{90\degree}{2}$

⇒ x° = 45°

Hence, x° = 45°.