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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 = 750,
∠AOC =300 then find the m∠BOC.

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

∴∠AOB = ∠AOC + ∠BOC

750 = 30 + ∠BOC

∴ ∠BOC = 75 - 30

∴ ∠BOC = 450

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3) The sum of two adjacent angle is 1100. If one angle is 300 more than the other. Find the measures of two angles.
Solution :
Let one angle be x.
Other angle = x + 30
Sum of two angles = 1100
∴ x + x + 30 = 110
⇒ 2x + 30 = 110
⇒ 2x = 110 - 30
⇒ 2x = 80
⇒ x = 80/2 =40
One angle = 400
So, other angle = x + 30
⇒ other angle = 40 + 30 = 700
∴ The two angles are 400 and 700.
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Basic Geometry

Point
Lines
Angles
Lines and Angles
Complementary angles
Supplementary angles
Vertically Opposite Angles
Linear Pair Angles