1. The problem asks why the angle in the third quadrant is expressed as $180^\circ + \alpha$ instead of $\alpha - 180^\circ$. 2. In trigonometry, angles are measured from the positive x-axis, moving counterclockwise. 3. The third quadrant is located between $180^\circ$ and $270^\circ$. 4. If $\alpha$ is the reference angle (the acute angle the terminal side makes with the x-axis), then the actual angle in the third quadrant is the sum of $180^\circ$ plus this reference angle. 5. Therefore, the angle is $180^\circ + \alpha$ because you start at $180^\circ$ (the negative x-axis) and move $\alpha$ degrees into the third quadrant. 6. Writing $\alpha - 180^\circ$ would give a negative or incorrect angle measure since $\alpha$ is acute and less than $90^\circ$. 7. Hence, the correct expression for an angle in the third quadrant is $180^\circ + \alpha$. Final answer: The third quadrant angle is $180^\circ + \alpha$ because angles are measured counterclockwise from the positive x-axis, and the third quadrant starts at $180^\circ$.