[geometry word problems][section 12]

(1.)  The measures of the angles of a triangle can be

      represented by s, (1/2)s, and 3s.   Find the measure

     of each angle of the triangle.

      s + (1/2)s + 3s = 180         here is the problem

       4.5s = 180            combine like terms
       ____  ____
      4.5    4.5           divide each side by 4.5

         s = 40           divide and cancel

     (1/2)s                 here is the problem

=   (1/2)(40)                 replace s with 40

=    20                      multiply

        3s                   here is the problem

=     3(40)             replace s with 40

=     120                 multiply

results:  The angles are 40, 20, and 120.

(2.)  The measure of one angle of a triangle is 70o .  The

       measure of each of the other angles can be represented

       by 5x.   Find the measures of the two angles.

     70 + 5x + 5x = 180              here is the problem

       10x + 70 = 180               combine like terms

            -70   -70    subtract 70 from each side
     ____________________
    10x        = 110              subtract
    ____         ____
    10           10             divide each side by 10

         x = 11                 divide and cancel

          5x                   here is the problem

=    5(11)                          replace x with 11

=     55                           multiply

results:  The angles are 70, 55 and 55.


(3.)   The measure of one angle of a triangle can be represented

       by 10p.   The measure of each of the other two angles is

       15 more than the first angle.   Find the measure of each

       angle of the triangle.

      10p + (10p + 15) + (10p + 15) = 180   here is the problem

       30p + 30 = 180             combine like terms

             -30  -30    subtract 30 from each side
     ____________________
       30p       = 150      subtract
       ____        ____
        30          30        divide each side by 30

           p = 5                 divide and cancel

      10p                       here is the problem

=    10(5)                          replace p with 5

=      50                          multiply

      10p + 15                     here is the problem

=    10(5) + 15                replace p with 5

=       65                     multiply & add

results:  The angles are 50, 65, and 65 .


(4.)   The sum of the measures of two angles is 90o .   The

        measure of one of these angles is 2r;  the measure of

        the other angle is 2r - 18.  Find the measure of each

        of these angles.

         2r + 2r - 18 = 90             here is the problem

             4r - 18 = 90             combine like terms

                  +18  +18          add 18 to each side
             _______________
              4r  =   108              add
              ___     ____
               4       4     divide each side by 4

                 r = 27       divide and cancel

        2r                     here is the problem

=     2(27)                     replace r with 27

=      54                        multiply

       2r - 18                     here is the problem

=   2(27) - 18               replace r with 27

=     36                        multiply and subtract

results:  the angles are 54 and 36.


(5.)    The sum two angles is 90o .  The measures of the angles

        can be represented by x and (x + 28).  Find the measure

         of each angle.

      x + x + 28 = 90              here is the problem

           2x + 28 = 90             combine like terms

               -28  -28       subtract 28 from each side
         ________________
          2x       = 62          subtract
         ___        ____
           2          2           divide each side by 2

            x = 31              divide and cancel

       x + 28                   here is the problem

=    31 + 28                    replace x with 31

=      59                        add

results:  the angles are 31 and 59.

(6.)     The sum of two angles is 90o .   The measures of the

         angles can be represented by t and (t - 12).  Find the

          measure of each angle.

      t + t - 12 = 90              here is the problem

       2t - 12 = 90             combine like terms

          + 12  +12           add 12 to each side
      _______________
        2t     = 102           add
        __      ____
         2        2          divide each side by 2

          t = 51            divide and cancel

      t - 12               here is the problem

=   51 - 12             replace t with 51

=    39                      subtract

results:  the angles are 51 and 39.


(7.)  The sum of two angles is 180o .  Their measures can

      be represented by y and (y + 36).  Find the measure of

      each angle.

       y + y + 36 = 180            here is the problem

        2y + 36 = 180          combine like terms

           -36   -36     subtract 36 from each side
      __________________
       2y      =  144    subtract
      ___        _____
        2         2       divide each side by 2

            y = 72          divide and cancel

      y + 36                 here is the problem

=   72 + 36            replace y with 72

=    108                add

results:  the angles are 72 and 108.

(8.)   The sum of two angles is 180o .   Their measures can be

       represented by w and (3w + 18).   Find the measure of

       each angle.

     w + 3w + 18 = 180             here is the problem

       4w + 18 = 180         combine like terms

            -18  -18     subtract 18 from each side
     _________________
       4w   =    162        subtract
       __      ______
       4         4        divide each side by 4

       w = 40.5           divide and cancel

       3w + 18              here is the problem

=    3(40.5) + 18            replace w with 40.5

=     121.5 + 18             multiply

=     139.5                  add

results:  the angles are 40.5 and 139.5


(9.)   The measures of the angles of a parallelogram can by

       represented by x, x, 3x, and 3x.   Find the measure of

       each angle of the parallelogram.

     x + x + 3x + 3x = 360         here is the problem

         8x = 360           combine like terms
         ___  ___
          8    8         divide each side by 8

         x = 45            divide and cancel

     3x                      here is the problem

=   3(45)              replace x with 45

=    135                 multiply

results:  the angles are 45, 45, 135, and 135

(10.)   The sum of the measures of two angles is 180o .  The

        measure of one angle is 5x.   The measure of the other

        angle is twice the first.  Find the measure of each

         angle.

         5x + 2(5x) = 180           here is the problem

         5x + 10x = 180             multiply

         15x = 180                 combine like terms
        _____ _____
        15     15           divide each side by 15

           x = 12           divide and cancel

           5x                  here is the problem

=         5(12)                 replace x with 12

=    60                        multiply

 results:  the angles are 60 and 120

(11.)   The sum of the measures of two angles is 90o .  The

         measure of one angle is 4q.  The measure of the second

       angle is three times the first.   Find the measure of

        each angle.

            4q + 3(4q) = 90        here is the problem

         4q + 12q = 90                multiply

            16q = 90            combine like terms
            ____  ___
            16    16       divide each side by 16

             q = 5.625   divide and cancel

       4q                  here is the problem

=      4(5.625)              replace q with 5.625

=      22.5                  multiply

      3(22.5)               here is the problem

=      67.5                 multiply

results:  the angles are 22.5 and 67.5

           


(12.)   The sum of the measures of two angles is 180o .  The

         measure of one angle is 7t.  The measure of the second

        angles is 2 more than the first.   Find the measure of

       each angle.

        7t + 7t + 2 = 180            here is the problem

          14t + 2 = 180              combine like terms

               -2   -2           subtract 2 from each side
         __________________
         14t        = 178           subtract
         ____         ____
          14           14      divide each side by 14

             t = 89/7       reduce and cancel

      7t                      here is the problem

=    7(89/7)                   replace t with 89/7

=     89                         cancel

     results:  the angles are 89 and 91 .