1. **Problem statement:** Given two parallel lines $m$ and $n$ cut by a transversal $p$, and angle 4 at the intersection of $p$ and $m$ measures $64^\circ$, find the measures of all numbered angles 1 through 8. 2. **Key properties:** - When a transversal crosses parallel lines, corresponding angles are equal. - Angles on a straight line sum to $180^\circ$. - Vertically opposite angles are equal. 3. **Find angle 3:** Angle 3 and angle 4 are on a straight line, so $$\text{angle 3} + \text{angle 4} = 180^\circ$$ $$\text{angle 3} + 64^\circ = 180^\circ$$ $$\text{angle 3} = 180^\circ - 64^\circ = 116^\circ$$ 4. **Find angle 1:** Angle 1 is vertically opposite to angle 4, so $$\text{angle 1} = 64^\circ$$ 5. **Find angle 2:** Angle 2 is vertically opposite to angle 3, so $$\text{angle 2} = 116^\circ$$ 6. **Find angles at line $n$ using corresponding angles:** - Angle 5 corresponds to angle 1, so $$\text{angle 5} = 64^\circ$$ - Angle 6 corresponds to angle 2, so $$\text{angle 6} = 116^\circ$$ 7. **Find angle 7:** Angle 7 and angle 6 are on a straight line, so $$\text{angle 7} + \text{angle 6} = 180^\circ$$ $$\text{angle 7} + 116^\circ = 180^\circ$$ $$\text{angle 7} = 64^\circ$$ 8. **Find angle 8:** Angle 8 is vertically opposite to angle 7, so $$\text{angle 8} = 64^\circ$$ **Final answers:** - Angle 1 = $64^\circ$ - Angle 2 = $116^\circ$ - Angle 3 = $116^\circ$ - Angle 4 = $64^\circ$ - Angle 5 = $64^\circ$ - Angle 6 = $116^\circ$ - Angle 7 = $64^\circ$ - Angle 8 = $64^\circ$