1. **State the problem:** We need to find the size of angle $c$ in a triangle where two sides are equal, and two angles are given as $30^\circ$ and $51^\circ$. 2. **Recall the properties:** In a triangle, the sum of all interior angles is always $180^\circ$. 3. **Identify the triangle type:** Since two sides are equal, the triangle is isosceles, meaning the angles opposite those sides are equal. 4. **Assign angles:** Let the two equal sides be opposite angles $30^\circ$ and $c$. Since these sides are equal, their opposite angles must be equal, so $c = 30^\circ$. 5. **Check the sum:** The three angles are $30^\circ$, $51^\circ$, and $c=30^\circ$. 6. **Sum the angles:** $$30^\circ + 51^\circ + 30^\circ = 111^\circ$$ which is less than $180^\circ$, so this contradicts the triangle angle sum rule. 7. **Reconsider angle assignments:** The two equal sides are opposite angles $c$ and the other angle, so the two equal angles must be equal. Given one angle is $51^\circ$, the other equal angle must be $c$. 8. **Set up equation:** $$c = 51^\circ$$ 9. **Sum angles:** $$30^\circ + 51^\circ + c = 180^\circ$$ 10. **Solve for $c$:** $$30 + 51 + c = 180$$ $$81 + c = 180$$ $$c = 180 - 81$$ $$c = 99^\circ$$ 11. **Conclusion:** The size of angle $c$ is $99^\circ$. **Final answer:** $c = 99^\circ$