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.