Mathematics

In figure, line $$l || m$$ and $$n$$ is a transversal. If $$\angle 1=40^o$$, find all the angles and check that all corresponding angles and alternate angles are equal.


SOLUTION
As per the diagram $$l||m$$ and $$n$$ is transversal,
$$ \angle 1 = 40^o$$

We know sum of angles on a straight line is $$180^o$$

$$\angle 2 + 40 = 180^o$$

$$\angle 4 + 40 = 180^o$$     
   
 => $$\angle 2  = 140^o$$ 

=>  $$\angle 4 = 140^o$$

We know that Vertically opposite angles are equal

$$ \angle 1 = \angle 3 = 40^o$$

$$ \angle 5 = \angle 7$$

$$ \angle 6 = \angle 8$$

We know that alternate interior angle are equal

$$ \angle 3 = \angle 5 = 40^o$$     [ From  $$\angle 3 = 40^o$$]

$$ \angle 4 = \angle 6 = 140^o$$      [ From  $$\angle 4 = 140^o$$]

Thus 

$$ \angle 5 = \angle 7 = 40^o$$        [ From  $$\angle 5 = 40^o$$]

$$ \angle 6 = \angle 8 = 140^o$$       [ From  $$\angle 6 = 140^o$$]

$$ \therefore$$ pair of corresponding angles are equal and alternate angles are equal

Corresponding angles

$$ \angle 1 =\angle 5 = 40^o$$

$$ \angle 2 =\angle 6 = 140^o$$

$$ \angle 3 =\angle 7 = 40^o$$

$$ \angle 4 =\angle 8 = 140^o$$

Alternate angles 

$$ \angle 1 = \angle 7 = 40^o$$ 

$$ \angle 2 = \angle 8 = 140^o$$ 
     
$$ \angle 3 = \angle 5 = 40^o$$   

$$ \angle 4 = \angle 6 = 140^o$$    
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