Question

All figures below show two intersecting lines. One of the four angles formed is given. Find the other three and write them in the figure.

Answer 1

(i) The given figure can be named as follows.

From the figure, we can observe that ∠AOC and ∠COB make a linear pair.

We know that the angles in a linear pair are supplementary.

∴ ∠AOC + ∠COB = 180°

Given ∠COB = 45°

⇒ ∠AOC + 45° = 180°

⇒ ∠AOC = 180° − 45° = 135°

Also, ∠AOC and ∠AOD make a linear pair.

∴ ∠AOC + ∠AOD = 180°

∠AOC = 135°, therefore

135° + ∠AOD = 180°

⇒ ∠AOD = 180° − 135° = 45°

Also, ∠AOD and ∠DOB make a linear pair.

∴ ∠AOD + ∠DOB = 180°

∠AOD = 45°, therefore

45° + ∠DOB = 180°

⇒ ∠DOB = 180° − 45° = 135°

The calculated angles can be marked in the given figure as follows.

(ii) The given figure can be named as follows.

From the figure, we can observe that ∠POR and ∠ROQ make a linear pair.

We know that the angles in a linear pair are supplementary.

∴ ∠POR + ∠ROQ = 180°

Given ∠ROQ = 120°

⇒ ∠POR + 120° = 180°

⇒ ∠POR = 180° − 120° = 60°

Also, ∠POR and ∠POS make a linear pair.

∴ ∠POR + ∠POS = 180°

∠POR = 60°, therefore

60° + ∠POS = 180°

⇒ ∠POS = 180° − 60° = 120°

Also, ∠POS and ∠SOQ make a linear pair.

∴ ∠POS + ∠SOQ = 180°

∠POS = 120°, therefore

120° + ∠SOQ = 180°

⇒ ∠SOQ = 180° − 120° = 60°

The calculated angles can be marked in the given figure as follows.

(iii) The given figure can be named as follows.

From the figure, we can observe that ∠POM and ∠MOQ make a linear pair.

We know that the angles in a linear pair are supplementary.

∴ ∠POM + ∠MOQ = 180°

Given ∠MOQ = 90°

⇒ ∠POM + 90° = 180°

⇒ ∠POM = 180° − 90° = 90°

Also, ∠MOQ and ∠QOL make a linear pair.

∴ ∠MOQ + ∠QOL = 180°

∠MOQ = 90°, therefore

90° + ∠QOL = 180°

⇒ ∠QOL = 180° − 90° = 90°

Also, ∠LOP and ∠POM make a linear pair.

∴ ∠LOP + ∠POM = 180°

∠POM = 90°, therefore

∠LOP + 90° = 180°

⇒ ∠LOP = 180° − 90° = 90°

The calculated angles can be marked in the given figure as follows.