A convex pentagon has interior angles with measures (5x-12), (2x+100), (4x+16), (6x+15), and (3x+41). What is x?

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The measures are degrees.

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Answer 1 · 2018-03-14T22:33:25

#x=19#.

The sum of the interior angles in any convex polygon with #n# sides is

#(n-2)xx180°#

So, for a pentagon (which has 5 sides), its interior angles sum to

#color(white)(=)(5-2)xx180°#
#=3xx180°#
#=540°#

We also know what each angle is, in terms of #x#. Thus, we can equate the sum of all 5 angles to the angle sum of #540°:#

#(5x"–"12)+(2x"+"100)+(4x"+"16)+(6x"+"15)+(3x"+"41)=540#

Combining like terms, we get

#20x + 160=540#

Now all we do is solve for #x#:

#20x=380#

#x = 19#

#ul("Double-check:")#

If #x=19#, then

#(5x"–"12)+(2x"+"100)+(4x"+"16)+(6x"+"15)+(3x"+"41)#

#=(95"–"12)+(38"+"100)+(76"+"16)+(114"+"15)+(57"+"41)#

#=83+138+92+129+98#

#=540#,

which is the answer we want.