RS AGGARWAL CLASS 9 CHAPTER 7 LINES AND ANGLES EXERCISE 7A

  EXERCISE 7A

PAGE NO-198

Question 1:

Define the following terms:
(i) Angle
(ii) Interior of an angle
(iii) Obtuse angle
(iv) Reflex angle
(v) Complementary angles
(vi) Supplementary angles

Answer 1:

(i) Two rays OA and OB, with a common end-point O, form an angle AOB that is represented as $\angle AOB$$\angle AOB$.


(ii) The interior of an angle is the set of all points in its plane, which lie on the same side of OA as B and also on the same side of OB as A.


(iii) An angle greater than $90°$$90°$ but less than $180°$$180°$ is called an obtuse angle.



(iv) An angle greater than $180°$$180°$ but less than $360°$$360°$ is called a reflex angle.



(v) Two angles are said to be complementary if the sum of their measures is $90°$$90°$.

(vi) Two angles are said to be supplementary if the sum of their measures is $180°$$180°$.

Question 2:

Find the complement of each of the following angles.
(i) 55°
(ii) 16°
(iii) 90°
(iv) $\frac{2}{3}$$\frac{2}{3}$ of a right angle

Answer 2:

Two angles whose sum is 90° are called complementary angles.

(i) Complement of 55° = 90° − 55° = 35°

(ii) Complement of $16°=\left(90-16\right)°$$16°=\left(90-16\right)°$$=74°$$=74°$

(iii) Complement of 90° = 90° − 90° = 0°

(iv) $\frac{2}{3}\text{of a right angle}\Rightarrow \left(90\times \frac{2}{3}\right)°=60°$$\frac{2}{3} \text{of a right angle}\Rightarrow \left(90\times \frac{2}{3}\right)°=60°$
$\text{Complement of}\frac{2}{3}\text{of a right angle}=\left(90-60\right)°=30°$$\text{Complement of }\frac{2}{3} \text{of a right angle}=\left(90-60\right)°=30°$

Question 3:

Find the supplement of each of the following angles.
(i) 42°
(ii) 90°
(iii) 124°
(iv) $\frac{3}{5}$$\frac{3}{5}$ of a right angle

Answer 3:

Two angles whose sum is 180° are called supplementary angles.

(i) Supplement of 42° = 180° − 42° = 138°

(ii) Supplement of 90° = 180° − 90° = 90°

(iii) Supplement of 124° = 180° − 124° = 56°

(iv) $\frac{3}{5}\text{of a right angle}\Rightarrow \left(\frac{3}{5}\times 90\right)=54°$$\frac{3}{5} \text{of a right angle}\Rightarrow \left(\frac{3}{5}\times 90\right)=54°$
Supplement of $\frac{3}{5}\text{of a right angle}=\left(180-54\right)°=126°$$\frac{3}{5} \text{of a right angle}=\left(180-54\right)°=126°$

Question 4:

Find the measure of an angle which is
(i) equal to its complement
(ii) equal to its supplement

Answer 4:

(i) Let the measure of the required angle be $x°$$x°$.
Then, in case of complementary angles:
$x+x=90°\Rightarrow 2x=90°\Rightarrow x=45°$$$\begin{gathered} x+x=90° \\ \Rightarrow 2x=90° \\ \Rightarrow x=45° \end{gathered}$$
Hence, measure of the angle that is equal to its complement is $45°$$45°$.

(ii) Let the measure of the required angle be $x°$$x°$​.
Then, in case of supplementary angles:
$x+x=180°\Rightarrow 2x=180°\Rightarrow x=90°$$$\begin{gathered} x+x=180° \\ \Rightarrow 2x=180° \\ \Rightarrow x=90° \end{gathered}$$
Hence, measure of the angle that is equal to its supplement is $90°$$90°$.

Question 5:

Find the measure of an angle which is 36° more than its complement.

Answer 5:

Let the measure of the required angle be $x°$$x°$.
Then, measure of its complement $=\left(90-x\right)°$$=\left(90-x\right)°$.
Therefore,
$x-\left(90°-x\right)=36°\Rightarrow 2x=126°\Rightarrow x=63°$$$\begin{gathered} x-\left(90°-x\right)=36° \\ \Rightarrow 2x=126° \\ \Rightarrow x=63° \end{gathered}$$
Hence, the measure of the required angle is $63°$$63°$.

Question 6:

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

Answer 6:

Let the measure of the angle be x°.

∴ Supplement of x° = 180° − x°

It is given that,

(180° − x°) − x° = 30°

⇒ 180° − 2x°= 30°

⇒ 2x° = 180° − 30° = 150°

⇒ x° = 75°

Thus, the measure of the angle is 75°.

Question 7:

Find the angle which is four times its complement.

Answer 7:

Let the measure of the required angle be $x$$x$.
Then, measure of its complement $=\left(90°-x\right)$$=\left(90°-x\right)$.
Therefore,
$x=\left(90°-x\right)4\Rightarrow x=360°-4x\Rightarrow 5x=360°\Rightarrow x=72°$$$\begin{gathered} x=\left(90°-x\right)4 \\ \Rightarrow x=360°-4x \\ \Rightarrow 5x=360° \\ \Rightarrow x=72° \end{gathered}$$
Hence, the measure of the required angle is $72°$$72°$.

Question 8:

Find the angle which is five times its supplement.

Answer 8:

Let the measure of the required angle be $x$$x$.
Then, measure of its supplement $=\left(180°-x\right)$$=\left(180°-x\right)$.
Therefore,
$x=\left(180°-x\right)5\Rightarrow x=900°-5x\Rightarrow 6x=900°\Rightarrow x=150°$$$\begin{gathered} x=\left(180°-x\right)5 \\ \Rightarrow x=900°-5x \\ \Rightarrow 6x=900° \\ \Rightarrow x=150° \end{gathered}$$
Hence, the measure of the required angle is $150°$$150°$.

Question 9:

Find the angle whose supplement is four times its complement.

Answer 9:

Let the measure of the required angle be $x°$$x°$.
Then, measure of its complement $=\left(90-x\right)°$$=\left(90-x\right)°$.
And, measure of its supplement$=\left(180-x\right)°$$=\left(180-x\right)°$.
Therefore,
$\left(180-x\right)=4\left(90-x\right)\Rightarrow 180-x=360-4x\Rightarrow 3x=180\Rightarrow x=60$$$\begin{gathered} \left(180-x\right)=4\left(90-x\right) \\ \Rightarrow 180-x=360-4x \\ \Rightarrow 3x=180 \\ \Rightarrow x=60 \end{gathered}$$
Hence, the measure of the required angle is $60°$$60°$.

Question 10:

Find the angle whose complement is one-third of its supplement.

Answer 10:

Let the measure of the required angle be $x°$$x°$.
Then, the measure of its complement $=\left(90-x\right)°$$=\left(90-x\right)°$.
And the measure of its supplement$=\left(180-x\right)°$$=\left(180-x\right)°$.
Therefore,
$\left(90-x\right)=\frac{1}{3}\left(180-x\right)\Rightarrow 3\left(90-x\right)=\left(180-x\right)\Rightarrow 270-3x=180-x\Rightarrow 2x=90\Rightarrow x=45$$$\begin{gathered} \left(90-x\right)=\frac{1}{3}\left(180-x\right) \\ \Rightarrow 3\left(90-x\right)=\left(180-x\right) \\ \Rightarrow 270-3x=180-x \\ \Rightarrow 2x=90 \\ \Rightarrow x=45 \end{gathered}$$
Hence, the measure of the required angle is $45°$$45°$.

Question 11:

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

Answer 11:

Let the two angles be 4x and 5x, respectively.
Then,
$4x+5x=90\Rightarrow 9x=90\Rightarrow x=10°$$$\begin{gathered} 4x+5x=90 \\ \Rightarrow 9x=90 \\ \Rightarrow x=10° \end{gathered}$$
Hence, the two angles are $4x=4\times 10°=40°\text{and}5x=5\times 10°=50°$$4x=4\times 10°=40° \text{and} 5x=5\times 10°=50°$.

Question 12:

Find the value of x for which the angles (2x – 5)° and (x – 10)° are the complementary angles.

Answer 12:

Two angles whose sum is 90° are called complementary angles.

It is given that the angles (2x – 5)° and (x – 10)° are the complementary angles.

∴ (2x – 5)° + (x – 10)° = 90°

⇒ 3x° – 15° = 90°

⇒ 3x° = 90° + 15° = 105°

⇒ x° = $\frac{105°}{3}$$\frac{105°}{3}$ = 35°

Thus, the value of x is 35.