# Adjacent angles

Adjacent angles : Two angles in a plane are called adjacent-angles, if

1) they have a common vertex,

2) they have a common arms
(rays) , and

3) their other arms lie on the opposite sides of the common arm.

In the above figure ∠ AOC and ∠ BOC have a common vertex O. Also they have a common arm OC and their arms are OA and OB.

So ∠AOC and ∠BOC are adjacent-angles.

∠AOB = ∠AOC + ∠BOC

Examples :

1) Write down each pair of adjacent-angles from the diagram given below :

Solution :
i) ∠AOB, ∠BOC (common ray is OB and O is a common vertex)

ii) ∠AOC, ∠COD (common ray is OC and O is a common vertex)

iii) ∠BOC, ∠COD(common ray is OC and O is a common vertex)

iv) ∠AOB , ∠BOD (common ray is OB and O is a common vertex)

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2)∠AOC and ∠BOC are adjacent-angles. If m∠AOB = 75
0 ,
∠AOC =30
0 then find the m∠BOC.

Solution :
As ∠AOC and ∠BOC are adjacent-angles.

∴∠AOB = ∠AOC + ∠BOC

75
0 = 30 + ∠BOC

∴ ∠BOC = 75 - 30

∴ ∠BOC = 45
0

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3) The sum of two adjacent angle is 110
0 . If one angle is 30 0 more than the other. Find the measures of two angles.
Solution :
Let one angle be x.
Other angle = x + 30
Sum of two angles = 110
0
∴ x + x + 30 = 110
⇒ 2x + 30 = 110
⇒ 2x = 110 - 30
⇒ 2x = 80
⇒ x = 80/2 =40
One angle = 40
0
So, other angle = x + 30
⇒ other angle = 40 + 30 = 70
0
∴ The two angles are 40
0 and 70 0 .
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