The sides of a triangle rule state that the sum of any two sides of a triangle has to be greater than the third side.
Chet using each option to determine which one is a possible value for x:
a. 5, 8, x=2
[tex]\begin{gathered} 5+8>2 \\ 13>2 \end{gathered}[/tex][tex]\begin{gathered} 8+2>5 \\ 10>5 \end{gathered}[/tex][tex]\begin{gathered} 2+5>8 \\ 7>8 \end{gathered}[/tex]The third sum does not apply to the triangle side rule, the sum of both sides is less than the length of the third one. x=2 is not a possible side length for this triangle.
b. 5, 8, x=4
[tex]\begin{gathered} 5+8>4 \\ 13>4 \end{gathered}[/tex][tex]\begin{gathered} 8+4>5 \\ 12>5 \end{gathered}[/tex][tex]\begin{gathered} 4+5>8 \\ 9>8 \end{gathered}[/tex]The sum of any two sides of the triangle is greater than the third one for every possible combination.
x=4 is a possible side length for this triangle.
c. 5, 8, x=14
[tex]\begin{gathered} 5+8>14 \\ 13>14 \end{gathered}[/tex]The sum of the side lengths is not greater than the length of the third side. This side length is not valid.
d. 5, 8, x=17
[tex]\begin{gathered} 5+8>17 \\ 13>17 \end{gathered}[/tex]The sum of the side lengths is not greater than the length of the third side. This side length is not valid.
