solve for m

Question

   if m∠1 =(4x +12)degrees, m∠2=(7x + 3)degrees, and m∠3 = 6y degrees

 

 

  then x = _______, m∠1 =__________, m∠2=_______ and y = _____

 

 

thanks

Answer 1

LO:  

Your comment problem description can be depicted as follows:

   \        /   

    \  m∠2  /

      \      /

        \    /  

m∠1   \ /   

        / \    m∠3

        /    \

      /      \

     /  m∠ 4 \  

   /          \

Given:

m∠1 = 4x + 12

m∠2 = 7x + 3

m∠3 = 6y

We can conclude that…

   m∠4 = 180 - m∠3……definition of a straight angle

  ∴m∠4 = 180 - 6y………substitution property

We now have the values of ALL angles in terms of “x and y.”  Recognize that we can use the definitions of straight angles (as above), supplementary angles, and vertical angles, to solve for “x and y” algebraically, and, hence, calculate the angle values. Review the definitions of each.

         m∠1 + m∠2 = 180……..definition of a straight angle

(4x + 12) + (7x + 3) = 180……..substitution property

            11x + 15 = 180……..simplify

                11x = 165……..subtract 15 from both sides

                ∴ x = 15….…...divide both sides by 11

Solve for values of m∠1 and m∠2 by substituting “x = 15”…..

     ∴m∠1 = 4(15) + 12 = 60 + 12 = 72°

     ∴m∠2 = 7(15) + 3   = 105 + 3 = 108°

By definition of vertical angles, we recognize…..

       m∠3 = m∠1…....definition of vertical angle

     ∴m∠3 = 72°……..substitution property

      m∠4 = m∠2 ……definition of vertical angle

    ∴m∠4 = 108°……substitution property

We can now solve for “y” knowing the value m∠4 = 108°……

m∠4 = 108°……..…calculated above

m∠4 = 180 - 6y……calculated above

m∠4 = m∠4……..…identity property

108 = 180 - 6y……substitution property

6y = 72…………..simplify, variables LHS, constants RHS

∴y = 12…….……divide both sides by 6

Check your work by adding values of any pair of adjacent angles and verify the total value satisfies the definitions supplementary angles and straight angles……

       m∠1 + m∠2 = 72 + 108 = 180……check ☑️

       m∠2 + m∠3 = 108 + 72 = 180……check ☑️

       m∠3 + m∠4 = 72 + 108 = 180……check ☑️

       m∠4 + m∠1 = 108 + 72 = 180……check ☑️

∴Our solution is correct. Always check your work. Answer summary….

x = 15

m∠1 = 72°

m∠2 = 108°

y = 12