Lines & Angles - Exercise 17(A)

Name each of the following angles :

(i)

(ii)

(iii)

Answer

(i) The vertex is R, so the angle is named with R in the middle.

∠PRQ

(ii) The vertex is Z, so the angle is named with Z in the middle.

∠YZX

(iii) The vertex is B, so the angle is named with B in the middle.

∠ABC

Classify each of the following marked angles as acute, right, obtuse, straight or a reflex angle :

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Answer

(i) The angle is greater than 90° but less than 180°.

∴ It is an obtuse angle.

(ii) The angle is exactly 90°.

∴ It is a right angle.

(iii) The angle is less than 90°.

∴ It is an acute angle.

(iv) The angle is greater than 180° but less than 360°.

∴ It is a reflex angle.

(v) The angle is greater than 90° but less than 180°.

∴ It is an obtuse angle.

(vi) The angle is greater than 180° but less than 360°.

∴ It is a reflex angle.

Classify each of the following as acute, obtuse or a reflex angle :

(i) 103°

(ii) 88°

(iii) 10°

(iv) 200°

(v) 89.5°

(vi) 105°

(vii) 176°

(viii) 182°

Answer

(i) 103°

The angle is more than 90° but less than 180°.

∴ It is an obtuse angle.

(ii) 88°

The angle is less than 90°.

∴ It is an acute angle.

(iii) 10°

The angle is less than 90°.

∴ It is an acute angle.

(iv) 200°

The angle is greater than 180° but less than 360°.

∴ It is a reflex angle.

(v) 89.5°

The angle is less than 90°.

∴ It is an acute angle.

(vi) 105°

The angle is more than 90° but less than 180°.

∴ It is an obtuse angle.

(vii) 176°

The angle is more than 90° but less than 180°.

∴ It is an obtuse angle.

(viii) 182°

The angle is greater than 180° but less than 360°.

∴ It is a reflex angle.

An angle is 30° larger than a straight angle. Find the angle.

Answer

A straight angle measures exactly 180°.

According to the problem, the required angle is 30° more than a straight angle.

∴ Required angle = 180° + 30° = 210°

The required angle is 210°.

At 5 o'clock the hands of a clock make an obtuse angle and a reflex angle. Find the measure of each of these angles.

Answer

A clock is a circle of 360°, divided into 12 hours. Therefore, each hour gap represents:

$\dfrac{360^\circ}{12} = 30^\circ$

Obtuse Angle:

At 5 o'clock, there are 5 hour gaps between the hands (from 12 to 5).

∴ 5 x 30° = 150°

Reflex Angle:

This is the larger outer angle between the two hands.

∴ 360° - 150° = 210°

The obtuse angle is 150° and the reflex angle is 210°.

At 10 o'clock the hands of a clock make an acute angle and a reflex angle. Find the measure of each of these angles.

Answer

A clock is a circle of 360°, divided into 12 hours. Therefore, each hour gap represents:

$\dfrac{360^\circ}{12} = 30^\circ$

Acute Angle:

At 10 o'clock, the minute hand is at 12 and the hour hand is at 10. The shortest distance between them is 2 hour gaps (from 10 to 11, and 11 to 12).

∴ 2 x 30° = 60°

Reflex Angle:

This is the larger outer angle between the two hands.

∴ 360° - 60° = 300°

The acute angle is 60° and the reflex angle is 300°.

How many degrees are there in :

(i) two right angles

(ii) $\dfrac{1}{6}$ right angle

(iii) $1\dfrac{1}{2}$ right angles

(iv) a straight angle

(v) $\dfrac{2}{5}$ straight angle

Answer

(i) two right angles

One right angle = 90°

∴ Two right angles = 2 x 90° = 180°

Hence, the answer is 180°.

(ii) $\dfrac{1}{6}$ right angle

Right angle = 90°

∴ $\dfrac{1}{6}$ right angle = $\dfrac{1}{6} \times 90^{\circ}$ = 15°

Hence, the answer is 15°.

(iii) $1\dfrac{1}{2}$ right angles

Let's convert mixed to improper fraction:

$1\dfrac{1}{2} = \dfrac{3}{2}$

Right angle = 90°

∴ $\dfrac{3}{2}$ right angles = $\dfrac{3}{2} \times 90^{\circ}$

= $\dfrac{3}{1} \times 45^{\circ}$

= 135°

Hence, the answer is 135°.

(iv) a straight angle

A straight angle is always 180°.

Hence, the answer is 180°.

(v) $\dfrac{2}{5}$ straight angle

Straight angle = 180°

∴ $\dfrac{2}{5}$ straight angle = $\dfrac{2}{5} \times 180^{\circ}$

= $\dfrac{2}{1} \times 36^{\circ}$

= 72°

Hence, the answer is 72°.

Write the complement of each of the following angles :

(i) 46°

(ii) 90°

(iii) $\dfrac{3}{5}$ of a right angle

(iv) (x + 12)°

(v) 34° 27'

(vi) 42° 36' 25"

Answer

(i) 46°

Complement of 46° = 90° − 46° = 44°

Hence, the answer is 44°.

(ii) 90°

Complement of 90° = 90° − 90° = 0°

Hence, the answer is 0°.

(iii) $\dfrac{3}{5}$ of a right angle

First, let's find the angle:

$\dfrac{3}{5}$ of a right angle = $\dfrac{3}{5} \times 90^{\circ}$

= $\dfrac{3}{1} \times 18^{\circ}$

= 54°

Complement of 54° = 90° − 54° = 36°

Hence, the answer is 36°.

(iv) (x + 12)°

Complement of (x + 12)° = 90° − (x + 12)°

= 90° − x° - 12°

= 78° - x°

= (78 - x)°

Hence, the answer is (78 - x)°.

(v) 34° 27'

To subtract 27' from 0', we borrow 1° from the 90° and convert it into 60'.

Complement of 34° 27' = 90° − 34° 27'

= 89° 60' − 34° 27'

Thus,

$\begin{array}{rcc} 89^\circ & 60' \\ -34^\circ & 27' \\ \hline 55^\circ & 33' \\ \hline \end{array}$

Complement of 34° 27' = 55° 33'

Hence, the answer is 55° 33'.

(vi) 42° 36' 25"

To subtract 25'' and 36' from 0'' and 0', we borrow 1° from the 90° to get 60', and then borrow 1' from that to get 60'', leaving 89° and 59'.

Complement of 42° 36' 25" = 90° − 42° 36' 25"

= 89° 59' 60" − 42° 36' 25"

Thus,

$\begin{array}{rrrc} 90° = 89^\circ & 59' & 60'' \\ -42^\circ & 36' & 25'' \\ \hline 47^\circ & 23' & 35'' \\ \hline \end{array}$

Complement of 42° 36' 25" = 47° 23' 35"

Hence, the answer is 47° 23' 35".

Write the supplement of each of the following angles :

(i) 58°

(ii) 105°

(iii) 0.6 of a right angle

(iv) (x - 30)°

(v) 62° 56'

(vi) 83° 45' 30"

Answer

(i) 58°

Supplement of 58° = 180° − 58° = 122°

Hence, the answer is 122°.

(ii) 105°

Supplement of 105° = 180° − 105° = 75°

Hence, the answer is 75°.

(iii) 0.6 of a right angle

First, let's calculate the angle:

Right angle = 90°

∴ 0.6 of a right angle = 0.6 x 90° = 54°

Supplement of 54° = 180° − 54° = 126°

Hence, the answer is 126°.

(iv) (x - 30)°

Supplement of (x - 30)° = 180° − (x - 30)°

= 180° − x° + 30°

= 210° - x°

= (210 - x)°

Hence, the answer is (210 - x)°.

(v) 62° 56'

To subtract 56' from 0', we borrow 1° from the 180° and convert it into 60'.

Supplement of 62° 56' = 180° - 62° 56'

= 179° 60' - 62° 56'

Thus,

$\begin{array}{rr} 179^\circ & 60' \\ -62^\circ & 56' \\ \hline 117^\circ & 04' \\ \hline \end{array}$

Supplement of 62° 56' = 117° 4'

Hence, the answer is 117° 4'.

(vi) 83° 45' 30"

To subtract 30'' and 45' from 00'' and 00', we borrow 1° from 180° to get 60', and then borrow 1' from that to get 60'', leaving 179° and 59'.

Supplement of 83° 45' 30" = 180° - 83° 45' 30"

= 179° 59' 60" - 83° 45' 30"

$\begin{array}{rrr} 179^\circ & 59' & 60'' \\ -83^\circ & 45' & 30'' \\ \hline 96^\circ & 14' & 30'' \\ \hline \end{array}$

Supplement of 83° 45' 30" = 96° 14' 30"

Hence, the answer is 96° 14' 30".

If x° and (x + 30)° are complements of each other, find the value of x.

Answer

Complementary angles sum = 90°

∴ x° + (x + 30)° = 90°

⇒ 2x° + 30° = 90°

⇒ 2x° = 90° - 30°

⇒ 2x° = 60°

⇒ x° = $\dfrac{60^{\circ}}{2}$

⇒ x° = 30°

Hence, the value of x is 30°.

If z° and (z + 50)° are supplements of each other, find the value of z.

Answer

Supplementary angles sum = 180°

∴ z° + (z + 50)° = 180°

⇒ 2z° + 50° = 180°

⇒ 2z° = 180° - 50°

⇒ 2z° = 130°

⇒ z° = $\dfrac{130^{\circ}}{2}$

⇒ z° = 65°

Hence, the value of z is 65°.

Two complementary angles are in the ratio 4 : 5. Find the angles.

Answer

Let angles be (4x)° and (5x)°

Complementary angles sum = 90°

∴ 4x° + 5x° = 90°

⇒ 9x° = 90°

⇒ x° = $\dfrac{90^{\circ}}{9}$

⇒ x° = 10°

Angles:

(4x)° = (4 x 10)° = 40°

(5x)° = (5 x 10)° = 50°

Hence, the required angles are 40° and 50°.

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

Answer

Let angles be (7x)° and (8x)°

Supplementary angles sum = 180°

∴ 7x° + 8x° = 180°

⇒ 15x° = 180°

⇒ x° = $\dfrac{180^{\circ}}{15}$

⇒ x° = 12°

Angles:

(7x)° = (7 x 12)° = 84°

(8x)° = (8 x 12)° = 96°

Hence, the required angles are 84° and 96°.

25% of an angle is the complement of 50°. Find the angle.

Answer

The complement of an angle is found by subtracting it from 90°.

∴ 90° − 50° = 40°

Let the required angle be x.

25% of x = 40°

$\Rightarrow \dfrac{25}{100} \times x = 40^{\circ} \\[1em] \Rightarrow \dfrac{1}{4} \times x = 40^{\circ} \\[1em] \Rightarrow x = 40^{\circ} \times 4 \\[1em] \Rightarrow x = 160^{\circ}$

Hence, the required angle is 160°.

80% of an angle is the supplement of 140°. Find the angle.

Answer

The supplement of an angle is found by subtracting it from 180°.

∴ 180° − 140° = 40°

Let the required angle be y.

80% of y = 40°

$\Rightarrow \dfrac{80}{100} \times y = 40^{\circ} \\[1em] \Rightarrow \dfrac{4}{5} \times y = 40^{\circ} \\[1em] \Rightarrow 4y = 40^{\circ} \times 5 \\[1em] \Rightarrow 4y = 200^{\circ} \\[1em] \Rightarrow y = \dfrac{200^{\circ}}{4} \\[1em] \Rightarrow y = 50^{\circ}$

Hence, the required angle is 50°.

An angle is double its complement. Find the angle.

Answer

Let the angle be x.

Its complement is (90° - x).

The problem states the angle is double its complement.

∴ x = 2(90° - x)

⇒ x = 180° - 2x

⇒ x + 2x = 180°

⇒ 3x = 180°

⇒ x = $\dfrac{180^{\circ}}{3}$

⇒ x = 60°

Hence, the required angle is 60°.

The measure of an angle is 20° more than its complement. Find the measure of the angle.

Answer

Let the angle be x.

Its complement is (90° - x).

The problem states the angle is 20° more than its complement.

∴ x = (90° - x) + 20°

⇒ x = 110° - x

⇒ x + x = 110°

⇒ 2x = 110°

⇒ x = $\dfrac{110^{\circ}}{2}$

⇒ x = 55°

Hence, the required angle measures 55°.

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

Answer

Let the angle be x.

Its supplement is (180° - x).

The problem states the angle is 30° less than its supplement.

∴ x = (180° - x) - 30°

⇒ x = 150° - x

⇒ x + x = 150°

⇒ 2x = 150°

⇒ x = $\dfrac{150^{\circ}}{2}$

⇒ x = 75°

Hence, the required angle measures 75°.

Find an angle which is two-thirds of its complement.

Answer

Let the angle be x.

Its complement is (90 - x).

The problem states the angle is $\dfrac{2}{3}$ of its complement.

∴ x = $\dfrac{2}{3}(90^{\circ} - x)$

⇒ 3x = 2(90° - x)

⇒ 3x = 180° - 2x

⇒ 3x + 2x = 180°

⇒ 5x = 180°

⇒ x = $\dfrac{180^{\circ}}{5}$

⇒ x = 36°

Hence, the required angle measures 36°.

Find an angle which is one-fifth of its supplement.

Answer

Let the angle be x.

Its supplement is (180° - x).

The problem states the angle is $\dfrac{1}{5}$ of its supplement.

∴ x = $\dfrac{1}{5}(180^{\circ} - x)$

⇒ 5x = 180° - x

⇒ 5x + x = 180°

⇒ 6x = 180°

⇒ x = $\dfrac{180^{\circ}}{6}$

⇒ x = 30°

Hence, the required angle measures 30°.

Find an angle which is 5° more than four times its supplement.

Answer

Let the angle be x.

Its supplement is (180° - x).

The problem states that the angle is 5° more than four times its supplement.

∴ x = 4(180° - x) + 5

⇒ x = 720° - 4x + 5

⇒ x = 725° - 4x

⇒ x + 4x = 725°

⇒ 5x = 725°

⇒ x = $\dfrac{725^{\circ}}{5}$

⇒ x = 145°

Hence, the required angle measures 145°.