1. **Problem:** Complete each statement based on definitions and theorems about angles. 2. **Definition of Congruence:** If $\angle D \cong \angle E$, then their measures are equal: $m\angle D = m\angle E$. 3. **Definition of Complementary Angles:** If $m\angle 1 + m\angle 2 = 90^\circ$, then $\angle 1$ and $\angle 2$ are complementary angles. 4. **Definition of Supplementary Angles:** If $\angle P$ and $\angle Q$ are supplementary angles, then $m\angle P + m\angle Q = 180^\circ$. 5. **Definition of a Right Angle:** If $m\angle JKL = 90^\circ$, then $\angle JKL$ is a right angle. 6. **Vertical Angles Theorem:** If $\angle 3$ and $\angle 4$ are vertical angles, then $\angle 3 \cong \angle 4$. 7. **Complement Theorem:** If $\angle S$ and $\angle T$ form a right angle, then $\angle S$ and $\angle T$ are complementary. 8. **Supplement (Linear Pair) Theorem:** If $\angle X$ and $\angle Y$ form a linear pair, then $\angle X$ and $\angle Y$ are supplementary. 9. **Congruent Complements Theorem:** If $\angle 1$ is complementary to $\angle 2$ and $\angle 2$ is complementary to $\angle 4$, then $\angle 1 \cong \angle 4$. 10. **Congruent Supplements Theorem:** If $\angle J$ is supplementary to $\angle K$ and $\angle J$ is supplementary to $\angle L$, then $\angle K \cong \angle L$. --- **Proof Completion:** Given: $\angle 1$ and $\angle 2$ form a linear pair; $\angle 1$ and $\angle 3$ are supplementary. Prove: $\angle 2 \cong \angle 3$. | Statements | Reasons | |-----------------------------------|---------------------------------| | 1. $\angle 1$ and $\angle 2$ form a linear pair | 1. Given | | 2. $\angle 1$ and $\angle 2$ are supplementary | 2. Definition of linear pair | | 3. $m\angle 1 + m\angle 2 = 180^\circ$ | 3. Definition of supplementary angles | | 4. $\angle 1$ and $\angle 3$ are supplementary | 4. Given | | 5. $m\angle 1 + m\angle 3 = 180^\circ$ | 5. Definition of supplementary angles | | 6. $m\angle 1 + m\angle 2 = m\angle 1 + m\angle 3$ | 6. Transitive property of equality | | 7. $m\angle 2 = m\angle 3$ | 7. Subtraction property of equality | | 8. $\angle 2 \cong \angle 3$ | 8. Definition of congruent angles | This completes the proof that $\angle 2$ is congruent to $\angle 3$.