ANGLES FORMED BY PARALLEL LINES AND TRANSVERSAL

A transversal is a line that intersects two lines in the same plane at two different points.

In the diagram shown below, let the lines l1 and l2 be parallel. Because the line m cuts the lines l1 and l2, the line m is transversal. 

So, the two parallel lines l1 and l2 cut by the transversal m. 

The angles formed by the two parallel lines l1 and l2 and the transversal min the above diagram will be as follows. 


Vertically opposite angles are equal.

m∠1  =  m3

m∠2  =  m4

m∠5  =  m7

m∠6  =  m8


Corresponding angles are equal.

m∠1  =  m5

m∠2  =  m6

m∠3  =  m7

m∠4  =  m8

Alternate interior  angles  are equal.

m∠3  =  m5

m∠4  =  m6

Alternate exterior angles  are equal.

m∠1  =  m7

m∠2  =  m8

Consecutive interior angles are supplementary.

m∠3 + m6  =  180°

m∠4 + m5  =  180°

Same side exterior angles are supplementary.

m∠1 + m8  =  180°

m∠2 + m7  =  180°

Example 1 :

In the figure given below,  let the lines l1 and l2 be parallel and m is transversal. If mF = 65°, find the measure of each of the remaining angles.  

Solution :

In the figure above, F and H are vertically opposite angles and they are equal.

m∠H  =  mF

m∠H  =  65°

H and D are corresponding angles and they are equal. 

m∠D  =  mH

m∠D  =  65°

D and B are vertically opposite angles and they are equal. 

m∠B  =  mD

m∠B  =  65°

F and E form a linear pair and they are supplementary. 

m∠F + mE  =  180°

Substitute mF  =  65°.

65° + mE  =  180°

m∠E  =  115°

E and G are vertically opposite angles and they are equal. 

m∠G  =  mE

m∠G  =  115°

G and C are corresponding angles and they are equal. 

m∠C  =  mG

m∠C  =  115°

C and A are vertically opposite angles and they are equal. 

m∠A  =  mC

m∠A  =  115°

Therefore, 

m∠A  =  mC  =  mE  =  mG  =  115°

m∠B  =  mD  =  mF  =  mH  =  65°

Example 2 :

In the figure given below,  let the lines l1 and lbe parallel and t is transversal. Find the value of x.

Solution :

In the figure above, (2x + 20)° and (3x - 10)° are corresponding angles and they are equal. 

(2x + 20)°  =  (3x - 10)°

2x + 20  =  3x - 10

30  =  x

Example 3 :

In the figure given below,  let the lines l1 and l2 be parallel and t is transversal. Find the value of x.

Solution :

In the figure above, (3x + 20)° and 2x° are consecutive interior angles and they are supplementary. 

(3x + 20)° + 2x°  =  180°

3x + 20 + 2x  =  180

5x + 20  =  180

5x  =  160

x  =  32

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