1. The problem asks: If a complex number lies in the third quadrant, then in which quadrant does its conjugate lie? 2. Recall that a complex number $z = x + yi$ lies in the third quadrant if both $x < 0$ and $y < 0$. 3. The conjugate of $z$ is $\overline{z} = x - yi$. 4. Since the conjugate changes the sign of the imaginary part, if $y < 0$ for $z$, then $-y > 0$ for $\overline{z}$. 5. Therefore, $\overline{z}$ has $x < 0$ and $y > 0$, which places it in the second quadrant. Final answer: The conjugate lies in the second quadrant.