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

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

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

∠AOC =30

As ∠AOC and ∠BOC are adjacent-angles.

∴∠AOB = ∠AOC + ∠BOC

75

∴ ∠BOC = 75 - 30

∴ ∠BOC = 45

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3) The sum of two adjacent angle is 110

Let one angle be x.

Other angle = x + 30

Sum of two angles = 110

∴ x + x + 30 = 110

⇒ 2x + 30 = 110

⇒ 2x = 110 - 30

⇒ 2x = 80

⇒ x = 80/2 =40

One angle = 40

So, other angle = x + 30

⇒ other angle = 40 + 30 = 70

∴ The two angles are 40

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