Question

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

The measures are degrees.

Answer 1

`x=19`.

Explanation:

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.