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Question

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

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Solution

(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°.


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