1. **State the problem:** We have two intersecting lines forming vertical and adjacent angles with measures $ (13x + 52)^\circ $, $ z^\circ $, and $ (7x + 82)^\circ $. We need to find the values of $ x $ and $ z $. 2. **Use angle relationships:** Vertical angles are equal, and adjacent angles on a straight line sum to 180°. 3. **Set up equations:** - Since $ (13x + 52)^\circ $ and $ z^\circ $ are vertical angles, they are equal: $$ z = 13x + 52 $$ - Since $ z^\circ $ and $ (7x + 82)^\circ $ are adjacent angles on a straight line, their sum is 180°: $$ z + (7x + 82) = 180 $$ 4. **Substitute $ z $ from the first equation into the second:** $$ (13x + 52) + (7x + 82) = 180 $$ 5. **Simplify and solve for $ x $:** $$ 13x + 52 + 7x + 82 = 180 $$ $$ 20x + 134 = 180 $$ $$ 20x = 180 - 134 $$ $$ 20x = 46 $$ $$ x = \frac{46}{20} $$ $$ x = \frac{\cancel{46}}{\cancel{20}} \text{ (simplify by dividing numerator and denominator by 2)} $$ $$ x = \frac{23}{10} = 2.3 $$ 6. **Find $ z $ using $ z = 13x + 52 $:** $$ z = 13(2.3) + 52 $$ $$ z = 29.9 + 52 $$ $$ z = 81.9 $$ **Final answers:** $$ x = 2.3 $$ $$ z = 81.9^\circ $$