1. **Problem Statement:** We have two triangles with sides $a$, $b$, and $c$. We need to fill in the blanks for the sums of two sides and compare them to the third side. 2. **Triangle Inequality Theorem:** For any triangle with sides $a$, $b$, and $c$, the sum of any two sides is always greater than the third side: $$a + b > c$$ $$b + c > a$$ $$c + a > b$$ 3. **Assuming two triangles:** - Triangle 1: $a=3$, $b=4$, $c=5$ - Triangle 2: $a=5$, $b=7$, $c=10$ 4. **Calculate sums for Triangle 1:** - $a + b = 3 + 4 = 7$ - $b + c = 4 + 5 = 9$ - $c + a = 5 + 3 = 8$ 5. **Compare sums to the third side for Triangle 1:** - $a + b = 7 > c = 5$ - $b + c = 9 > a = 3$ - $c + a = 8 > b = 4$ 6. **Calculate sums for Triangle 2:** - $a + b = 5 + 7 = 12$ - $b + c = 7 + 10 = 17$ - $c + a = 10 + 5 = 15$ 7. **Compare sums to the third side for Triangle 2:** - $a + b = 12 > c = 10$ - $b + c = 17 > a = 5$ - $c + a = 15 > b = 7$ **Final answers:** - $a + b = 7$, and $a + b > c$ - $b + c = 9$, and $b + c > a$ - $c + a = 8$, and $c + a > b$ - $a + b = 12$, and $a + b > c$ - $b + c = 17$, and $b + c > a$ - $c + a = 15$, and $c + a > b$