1. The problem asks to find the range of possible measures for the third circle, $c$, given two values $a$ and $b$, using the inequality: $$|a-b| < c < a+b$$ 2. This inequality means $c$ must be greater than the absolute difference of $a$ and $b$, and less than their sum. 3. We will apply this formula to each pair given in the answer section. 4. For each pair $(a,b)$: - Calculate $|a-b|$ - Calculate $a+b$ - Write the range $|a-b| < c < a+b$ 5. Now, solve each: **1> (7,12):** $$|7-12| = 5$$ $$7+12 = 19$$ Range: $$5 < c < 19$$ **2> (8,15):** $$|8-15| = 7$$ $$8+15 = 23$$ Range: $$7 < c < 23$$ **3> (47,21):** $$|47-21| = 26$$ $$47+21 = 68$$ Range: $$26 < c < 68$$ **4> (30,19):** $$|30-19| = 11$$ $$30+19 = 49$$ Range: $$11 < c < 49$$ **5> (11,3):** $$|11-3| = 8$$ $$11+3 = 14$$ Range: $$8 < c < 14$$ 6. These ranges represent the possible measures for the third circle $c$ for each pair of $a$ and $b$. Final answers: 1. $$5 < c < 19$$ 2. $$7 < c < 23$$ 3. $$26 < c < 68$$ 4. $$11 < c < 49$$ 5. $$8 < c < 14$$