1. **Problem Statement:** We are given two parallel lines $l$ and $m$ intersected by a transversal, forming eight angles. Angle 2 measures $(3x + 70)^\circ$ and angle 5 measures $(x^2 + 70)^\circ$. We need to find the measure of angle 5. 2. **Key Concept:** When two parallel lines are cut by a transversal, corresponding angles are equal. Angle 2 and angle 5 are corresponding angles because they are in the same relative position at each intersection. 3. **Set up the equation:** Since angle 2 and angle 5 are corresponding angles, $$3x + 70 = x^2 + 70$$ 4. **Simplify the equation:** Subtract 70 from both sides, $$3x = x^2$$ 5. **Rewrite the equation:** $$x^2 - 3x = 0$$ 6. **Factor the equation:** $$x(x - 3) = 0$$ 7. **Solve for $x$:** $$x = 0 \quad \text{or} \quad x = 3$$ 8. **Check for valid solution:** If $x=0$, angle 2 becomes $3(0)+70=70^\circ$ and angle 5 becomes $0^2+70=70^\circ$, which is valid. If $x=3$, angle 2 is $3(3)+70=79^\circ$ and angle 5 is $3^2+70=79^\circ$, also valid. 9. **Find measure of angle 5:** Using $x=3$ (non-zero value), $$\angle 5 = x^2 + 70 = 3^2 + 70 = 9 + 70 = 79^\circ$$ **Final answer:** The measure of angle 5 is $79^\circ$.