[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

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
_______________
___     ____
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

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
_______________
__      ____
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

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

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 .