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Theorems on Linear Pair and Vertically Opposite Angles

Linear Pair Axiom :-
Axiom1
If a ray stands on a line , then the sum of two adjacent angles so formed is 180
°

 

Axiom2
If the sum of two adjacent angles is 180
° then the non common arms of the angles form a line.

Consider the angles formed by rays OA, OB, OC and OD, having a common end point O. Then the four angles together form a complete angle, i.e.

1 + 2 + 3 + 4 = 360°


In general, the sum of all the angles at a point is 360° or 4 right angles.
 

Example : Find the angles x, y and z in the figure.

∠ x + 130° = 180° (Linear pair)

Therefore ∠ x = 180° - 130° = 50°

∠ y = 50° (since x and y are vertically opposite angles, ∠ x = ∠ y)

∠ z =180° - ∠ y (Linear pair)

= 180°-50°= 130°

 

Example : Find the angles in the figure.
 

x + 2x - 30° = 90° (Complementary angles)

3x = 120° or x =40°

Therefore 2x - 30° = 80° - 30° = 50°

The angles are 40° and 50°.


Theorem 1
If two lines intersect each other, then vertically opposite angles are equal.
Given: Two lines AB and CD intersect each other at O.
To Prove: AOD = COB and AOC = DOB
 


Proof: Ray OD stands on the line AB
AOD + DOB = 180° ….(i)
Since ray OA stands on the line CD
COA + AOD = 180° …. (ii)
Comparing equation (i) and (ii), we get

AOD + DOB = COA + AOD
DOB = COA
Similarly we can prove that

AOD = COB
 

Example :

In the figure, lines l1 and l2 intersect at O, forming angles as shown. If x = 53°, find the values of y, z and p.

Solution

Since l1 and l2 intersect each other
 
... ∠x = z (Vertically opposite angles)
... 53° = z
...  z = 53°
Since y and z form a linear pair

... ∠y + z = 180°
...∠y + 53° = 180°
...∠y = 180° - 53° = 127°
Again,
y = p (Vertically opposite angles)
...  127° = p or p = 127°
Hence x = z = 53° and y = p = 127o

 





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