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# Provided Answer - Wanted Solution

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What is the value of B in the picture?

Guest Aug 3, 2017

#1
+4695
+3

Here is one way...

These angles form a straight line, so they sum to 180°  .

b + c + d  =  180                   Since  c  and  d  are vertical angles,  c = d .

b + d + d  =  180

b + 2d  =  180

2d  =  180 - b                        Divide through by  2  .

d  =  90 - b/2

These angles form a straight line, so they sum to 180° .

a + d + (a - b)  =  180            Since  a  and  d  are vertical angles,  a = d .

d + d + (d - b)  =  180

d + d + d - b  =  180

3d - b  =  180                       Substitute  90 - b/2  in for  d .

3(90 - b/2) - b  =  180           Distribute the  3  .

270 - 3b/2 - b  =  180           Subtract  270  from both sides.

-3b/2 - b  =  -90                    Multiply through by  2  .

-3b - 2b  =  -180

-5b  =  -180                          Divide both sides by  -5  .

b  =  36

hectictar  Aug 3, 2017
Sort:

#1
+4695
+3

Here is one way...

These angles form a straight line, so they sum to 180°  .

b + c + d  =  180                   Since  c  and  d  are vertical angles,  c = d .

b + d + d  =  180

b + 2d  =  180

2d  =  180 - b                        Divide through by  2  .

d  =  90 - b/2

These angles form a straight line, so they sum to 180° .

a + d + (a - b)  =  180            Since  a  and  d  are vertical angles,  a = d .

d + d + (d - b)  =  180

d + d + d - b  =  180

3d - b  =  180                       Substitute  90 - b/2  in for  d .

3(90 - b/2) - b  =  180           Distribute the  3  .

270 - 3b/2 - b  =  180           Subtract  270  from both sides.

-3b/2 - b  =  -90                    Multiply through by  2  .

-3b - 2b  =  -180

-5b  =  -180                          Divide both sides by  -5  .

b  =  36

hectictar  Aug 3, 2017
#2
+76833
+2

Thanks, hectictar...here's one more way...

Note that b and  (a - b)   are vretical angles so they equal each other

Therefore....     b = a - b  →    a  = 2b

And c and  d  are vertical angles...so.... c  = d

So we actually have this system

b + c + d  = 180

a + d + (a - b)  = 180   substituting, we have that

b + c + c   = 180

2b + c + (2b - b) = 180       simplify

b + 2c  = 180

3b + c = 180        multiply the second  equation by -2

b + 2c  =  180

-6b - 2c  = -360     add these

-5b  = -180        divide both sides by  -5

b  = 36

CPhill  Aug 3, 2017
#3
+4695
+3

I think I figured out a better way still....

Since  a = d = c  , we can replace  c  and  d  with  a  .

And...

b + a + a + (a - b) + a + a  =  360°

b + a + a + a - b + a + a  -  360°

5a  =  360°

a  =  72°

a - b  =  b

a  =  2b

72°  =  2b

36°  =  b

hectictar  Aug 4, 2017
#4
+76833
+1

Yep....that makes it way easier.....!!!!

CPhill  Aug 4, 2017

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