AUXILIARY LINE

Auxiliary line (Helping line) is an extra line needed to complete a proof or problem in plane geometry. 

In the figure above, dotted line is an auxiliary line. Through any point there is exactly one line that will be parallel to an already existing line.   

Proving the Triangle Sum Theorem Using an Auxiliary Line

Given : ΔABC

Prove : m∠1 + m∠2 + m∠3  =  180°

Recall : The sum of the interior angles of a triangle 180°.

Plane for the proof : 

Draw an auxiliary line through B that is parallel to AC, name the new angles on this line m∠4 and m∠5.

Demonstrate that m∠4 + m∠2 + m∠5 = 180°, 

m∠1 ≅ m∠4  and  m∠3 ≅ m∠5

Use substitution to get 

m∠1 + m∠2 + m∠3  =  180°

Solved Problems

Problem 1 :

Using a 3^rd parallel Line – Auxiliary Line, find the value of x. 

Solution :

In the figure above, a° and 50° are corresponding angles and they are equal.

a°  =  50°

b° and 105° are interior angles on the same side of the transversal and they are supplementary. 

b° + 100°  =  180°

Subtract 100° from each side. 

b°  =  80°

In the above figure, 

x  =  a + b

=  50 + 80

=  130

Problem 2 :

Using a 3^rd parallel Line – Auxiliary Line, find the value of x. 

Solution :

In the figure above, a° and 60° are alternate interior angles and they are equal.

a°  =  62°

b° and 144° are interior angles on the same side of the transversal and they are supplementary. 

b° + 144°  =  180°

Subtract 144° from each side. 

b°  =  36°

In the above figure,

x  =  a + b

=  62 + 36

=  98

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here. 

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.