Example 107.

The second angle of a triangle is double the first. The third angle is \(40\) less than the first. Find the three angles.

First \(x\) With nothing given about the first we make that \(x\)
Second \(2x\) The second is double the first,
Third \(x − 40\) The third is \(40\) less than the first
\(F + S + T = 180\) All three angles add to \(180\)
\((x) +(2x)+ (x − 40)= 180\) Replace \(F\) , \(S\), and \(T\) with the labeled values
\(x +2x + x − 40 = 180\) Here the parenthesis are not needed.
\(4x − 40 = 180\) Combine like terms, \(x +2x + x\)
\(\underline {+ 40 + 40}\) Add \(40\) to both sides
\(\underline {4x = 220} \) The variable is multiplied by \(4\)
\(4 \quad \quad 4\) Divide both sides by \(4\)
\(x = 55\) Our solution for \(x\)
First \(55\) Replace \(x\) with \(55\) in the original list of angles
Second \(2(55)= 110\)
Third \((55) − 40 = 15\)
 Our angles are \(55\), \(110\), and \(15\)


Another geometry problem involves perimeter or the distance around an object. For example, consider a rectangle has a length of 8 and a width of 3. There are two lengths and two widths in a rectangle (opposite sides) so we add \(8 + 8 + 3 + 3 = 22\). As there are two lengths and two widths in a rectangle an alternative to find the perimeter of a rectangle is to use the formula \(P = 2L + 2W\). So for the rectangle of length 8 and width 3 the formula would give, \(P = 2(8) + 2(3) = 16 + 6 = 22\). With problems that we will consider here the formula \(P = 2L + 2W\) will be used.