1. **State the problem:** We have three line segments originating from a single point forming two angles: one angle is $(3z+2)^\circ$ and the other is $(8z)^\circ$. There is a right angle (90°) between the vertical and horizontal lines. 2. **Understand the geometry:** The sum of the angles around a point on a straight line is 180°. Since the vertical and horizontal lines form a right angle (90°), the other two angles must add up to the remaining 90°. 3. **Set up the equation:** $$ (3z+2) + 8z = 90 $$ 4. **Simplify the equation:** $$ 3z + 2 + 8z = 90 $$ $$ 11z + 2 = 90 $$ 5. **Isolate $z$:** $$ 11z = 90 - 2 $$ $$ 11z = 88 $$ 6. **Divide both sides by 11:** $$ z = \frac{\cancel{11}z}{\cancel{11}} = \frac{88}{11} $$ 7. **Calculate the value:** $$ z = 8 $$ 8. **Verify by substituting back:** $$ (3z+2) = 3(8) + 2 = 24 + 2 = 26^\circ $$ $$ (8z) = 8(8) = 64^\circ $$ $$ 26^\circ + 64^\circ = 90^\circ $$ which matches the right angle. **Final answer:** $$ z = 8 $$