Question

In the figures below, find the measures of the angles marked.

Answer 1

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

From the figure, we can observe that ∠AOF and ∠BOE, ∠AOC and ∠DOB and ∠FOD and ∠COE are opposite angles.

We know that the opposite angles between two lines are equal.

So, we have

∠DOB = ∠AOC = 70° and ∠COE = ∠DOF = 30°

Also, ∠AOF, ∠FOD and ∠DOB make a linear set.

∴ ∠AOF + ∠FOD + ∠DOB = 180°

⇒ ∠AOF + 30° + 70° = 180°

⇒ ∠AOF + 100° = 180°

⇒ ∠AOF = 180° − 100° = 80°

Also, ∠BOE = ∠AOF = 80°

Thus, we have ∠AOF = 80°, ∠DOB = 70°, ∠BOE = 80° and ∠COE = 30°.

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

From the figure, we can observe that ∠AOE and ∠BOF, ∠DOE and ∠COF, and ∠AOC and ∠DOB are opposite angles.

We know that the opposite angles between two lines are equal.

So, we have

∠AOE = ∠BOF = 60° and ∠AOC = ∠DOB = 60°

Also, ∠AOE, ∠AOC and ∠COF make a linear set.

∴ ∠AOE + ∠AOC + ∠COF = 180°

⇒ 60° + 60° + ∠COF = 180°

⇒ 120° + ∠COF = 180°

⇒ ∠COF = 180° − 120° = 60°

Also, ∠DOE = ∠COF = 60°

Thus, we have ∠AOC = ∠COF = ∠BOF = ∠DOE = 60°.

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

From the figure, we can observe that ∠AOX and ∠BOY, as well as ∠AOY and ∠BOX, are opposite angles.

We know that the opposite angles between two lines are equal.

So, we have

∠AOX = ∠BOY = 110° and ∠AOY = ∠BOX

Also, ∠BOY = ∠BOC + ∠COY

⇒ 110° = 60° + ∠COY

⇒ ∠COY = 110° − 60° = 50°

Also, ∠AOX and ∠AOY make a linear pair.

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

∴ ∠AOX + ∠AOY = 180°

⇒ 110° + ∠AOY = 180°

⇒ ∠AOY = 180° − 110° = 70°

So, we have ∠AOY = ∠BOX = 70°.

Thus, we have ∠AOY = ∠BOX = 70° and ∠COY = 50°.

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

From the figure, we can observe that ∠AOC and ∠BOD, as well as ∠AOD and ∠BOC, are opposite angles.

We know that the opposite angles between two lines are equal.

So, we have

∠AOC = ∠BOD = 130° and ∠AOD = ∠BOC

Also, ∠AOC = ∠AOE + ∠EOC

⇒ 130° = ∠AOE + 50°

⇒ ∠AOE = 130° − 50° = 80°

Also, ∠AOE and ∠EOB make a linear pair.

∴ ∠AOE + ∠EOB = 180°

⇒ ∠AOE + ∠EOC + ∠COB = 180°

⇒ 80° + 50° + ∠COB = 180°

⇒ 130° + ∠COB = 180°

⇒ ∠COB = 180° − 130° = 50°

So, we have ∠COB = ∠AOD = 50°.

Thus, we have ∠AOE = 80° and ∠COB = ∠AOD = 50°.