1. The problem asks us to determine side lengths that satisfy the triangle inequality theorem. 2. The triangle inequality theorem states that for any triangle with sides $a$, $b$, and $c$, the following must be true: $$a + b > c$$ $$a + c > b$$ $$b + c > a$$ This means the sum of the lengths of any two sides must be greater than the length of the remaining side. 3. For the isosceles triangle (two sides equal), suppose the equal sides are each length $x$ and the base is length $y$. The inequalities are: $$x + x > y \Rightarrow 2x > y$$ $$x + y > x \Rightarrow y > 0$$ $$x + y > x \Rightarrow y > 0$$ 4. For example, if we use the side lengths from class: $x = 5$, $y = 8$, check: $$2 \times 5 = 10 > 8$$ (True) $$5 + 8 = 13 > 5$$ (True) $$5 + 8 = 13 > 5$$ (True) All inequalities hold, so these side lengths satisfy the theorem. 5. For the scalene triangle (all sides different), suppose sides are $a=4$, $b=6$, and $c=9$. Check: $$4 + 6 = 10 > 9$$ (True) $$4 + 9 = 13 > 6$$ (True) $$6 + 9 = 15 > 4$$ (True) All inequalities hold, so these side lengths satisfy the theorem. Final answer: Side lengths $5, 5, 8$ for the isosceles triangle and $4, 6, 9$ for the scalene triangle satisfy the triangle inequality theorem.