In this figure, m∠3 = (2x + 15)° and m∠6 = (3x + 10)°. what value of x makes a∥b?

 " x = 31 " .


Explanation:


If the two lines that are shown in the figure:  "line a" and "line b" ;  are parallel;{ that is;  if  " a∥b " } :
 
{ that is;  if  " a∥b " } :

The following would be true:
 
1)  "∠2 & ∠8 " would be equal; since they are "alternate exterior angles".
2)  "∠1 & ∠7 " would be equal; since they are "alternate exterior angles".

3) "∠1 & ∠3" would be equal; since they are "vertical angles". 

4) "∠5 & ∠7 would be equal; since they are "vertical angles".  

5) "∠8 & ∠6 would be equal; since they are "vertical angles".


So, if  " a∥b " ;  then:
 
  "  m∠1 = m∠3 = m∠5 = m∠7 " .
 and:  " m∠2 = m∠4 = m∠6 = m∠8 " ; 

and:  " m∠6 + m∠3 = 180 " ; 

since:  "m∠3 = m∠7 " ; 

and:  "∠5 & ∠7" are "supplementary angles" ; which, by definition, add up to 180.
 
Also, look at "∠2 and ∠3" . These are "supplementary angles" , which, by definition, add up to 180.
 
We know that "m∠2 = m∠6"  ;  So:  " m∠2  + m∠3 = m∠6 + m∠3 = 180 ".

Given "m∠3 + m∠6 = 180" ; 

and given:  "m∠3 = 2x + 15" ; 

and given:  "m∠6 = 3x + 10" ;


→ 2x + 15 + 3x + 10  = 180 ;  Solve for "x" ;

→  2x + 3x = 5x ;

→  10 + 15 = 25 ;


→  5x + 25 = 180 ; 

Subtract "25" from each side of the equation:

→  5x + 25 − 25 = 180 − 25 ; 

to get:

→  5x = 155 ; 

Divide each side of the equation by "5" ; 
   to isolate "x" on one side of the equation; & to solve for "x" ; 

→  5x / 5 = 155 / 5 ; 

to get:

→  " x = 31 " .