1. **State the problem:** We are given three angle sum expressions involving angles of triangle ABC and angle CAD. We want to understand and verify these expressions. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always 180 degrees. This means: $$m\angle ABC + m\angle BAC + m\angle BCA = 180^\circ$$ 3. **Analyze the second expression:** $$m\angle BAC = 180^\circ - m\angle CAD$$ This suggests that angle BAC and angle CAD are supplementary (sum to 180 degrees). 4. **Substitute the second expression into the first:** Replace $m\angle BAC$ in the first equation with $180^\circ - m\angle CAD$: $$m\angle ABC + (180^\circ - m\angle CAD) + m\angle BCA = 180^\circ$$ 5. **Simplify the third expression:** Combine like terms: $$m\angle ABC + 180^\circ - m\angle CAD + m\angle BCA = 180^\circ$$ Subtract $180^\circ$ from both sides: $$m\angle ABC + \cancel{180^\circ} - m\angle CAD + m\angle BCA = \cancel{180^\circ}$$ Which simplifies to: $$m\angle ABC - m\angle CAD + m\angle BCA = 0$$ 6. **Interpretation:** This final equation shows a relationship between angles ABC, BCA, and CAD. It implies that the sum of angles ABC and BCA equals angle CAD: $$m\angle ABC + m\angle BCA = m\angle CAD$$ This is consistent with the idea that angle CAD is an exterior angle to triangle ABC at vertex A. **Final answer:** $$m\angle ABC + m\angle BCA = m\angle CAD$$