1. **Problem:** Find the length of side BC in triangle ABC where AB = AC = 24 and BD = DC = 15. 2. **Step 1:** Recognize that triangle ABC is isosceles with AB = AC = 24. 3. **Step 2:** Since BD = DC = 15, point D is the midpoint of BC, so BC = BD + DC = 15 + 15 = 30. 4. **Step 3:** Therefore, BC = 30. **Final answer:** BC = 30 --- 2. **Problem:** Find AB in triangle ABC where AB = 4x + 6, AC = 5x - 6, and AB = AC (since triangle is isosceles). 3. **Step 1:** Set the expressions equal: $$4x + 6 = 5x - 6$$ 4. **Step 2:** Solve for $x$: $$4x + 6 = 5x - 6$$ $$6 + 6 = 5x - 4x$$ $$12 = x$$ 5. **Step 3:** Substitute $x = 12$ back into $AB = 4x + 6$: $$AB = 4(12) + 6 = 48 + 6 = 54$$ **Final answer:** AB = 54 --- 3. **Problem:** Find $m\angle BAD$ given triangle ABD with sides BD = DC = 7 and angle at A is 23°. 4. **Step 1:** Since BD = DC, triangle BDC is isosceles with base BC. 5. **Step 2:** The angle at A is given as 23°, and since BD = DC, angles BAD and CAD are equal. 6. **Step 3:** Therefore, $m\angle BAD = 23^\circ \times 2 = 46^\circ$. **Final answer:** $m\angle BAD = 46^\circ$