1. **Problem:** Find $m\angle 1$ if $\angle 1$ and $\angle 2$ are vertical angles and $m\angle 2 = 112^\circ$. 2. **Formula and rule:** Vertical angles are congruent, so $m\angle 1 = m\angle 2$. 3. **Calculation:** $$m\angle 1 = 112^\circ$$ 4. **Answer:** $m\angle 1 = 112^\circ$. 1. **Problem:** Find $m\angle D$ if $\angle C$ and $\angle D$ are complementary and $m\angle C = 13^\circ$. 2. **Formula and rule:** Complementary angles sum to $90^\circ$, so $$m\angle C + m\angle D = 90^\circ$$ 3. **Calculation:** $$13^\circ + m\angle D = 90^\circ$$ $$m\angle D = 90^\circ - 13^\circ$$ $$m\angle D = 77^\circ$$ 4. **Answer:** $m\angle D = 77^\circ$. 1. **Problem:** Find $m\angle B$ if $\angle 4$ and $\angle B$ are supplementary and $m\angle 4 = 89^\circ$. 2. **Formula and rule:** Supplementary angles sum to $180^\circ$, so $$m\angle 4 + m\angle B = 180^\circ$$ 3. **Calculation:** $$89^\circ + m\angle B = 180^\circ$$ $$m\angle B = 180^\circ - 89^\circ$$ $$m\angle B = 91^\circ$$ 4. **Answer:** $m\angle B = 91^\circ$. 1. **Problem:** Find $m\angle 4$ if $\angle 3$ and $\angle 4$ are vertical angles and $m\angle 3 = 47^\circ$. 2. **Formula and rule:** Vertical angles are congruent, so $$m\angle 4 = m\angle 3$$ 3. **Calculation:** $$m\angle 4 = 47^\circ$$ 4. **Answer:** $m\angle 4 = 47^\circ$. 1. **Problem:** Find $m\angle N$ if $\angle M$ and $\angle N$ are complementary and $m\angle M = 6^\circ$. 2. **Formula and rule:** Complementary angles sum to $90^\circ$, so $$m\angle M + m\angle N = 90^\circ$$ 3. **Calculation:** $$6^\circ + m\angle N = 90^\circ$$ $$m\angle N = 90^\circ - 6^\circ$$ $$m\angle N = 84^\circ$$ 4. **Answer:** $m\angle N = 84^\circ$. 1. **Problem:** Find $m\angle R$ if $\angle R$ and $\angle S$ are supplementary and $m\angle S = 12^\circ$. 2. **Formula and rule:** Supplementary angles sum to $180^\circ$, so $$m\angle R + m\angle S = 180^\circ$$ 3. **Calculation:** $$m\angle R + 12^\circ = 180^\circ$$ $$m\angle R = 180^\circ - 12^\circ$$ $$m\angle R = 168^\circ$$ 4. **Answer:** $m\angle R = 168^\circ$.