1. **State the problem:** We have an isosceles trapezoid with sides AB = 3x - 2, BC = 5x - 6, CD = 3x + 9, and AD = 4x + 5. We need to find the value of $x$ and then find $m\angle A$ given $m\angle C = 72^\circ$. 2. **Use properties of isosceles trapezoids:** In an isosceles trapezoid, the non-parallel sides (legs) are equal in length. Here, the legs are AD and BC, so: $$4x + 5 = 5x - 6$$ 3. **Solve for $x$:** $$4x + 5 = 5x - 6$$ Subtract $4x$ from both sides: $$\cancel{4x} + 5 = \cancel{4x} + 5x - 6 \Rightarrow 5 = x - 6$$ Add 6 to both sides: $$5 + 6 = x - 6 + 6 \Rightarrow 11 = x$$ 4. **Find the lengths of the bases:** $$AB = 3x - 2 = 3(11) - 2 = 33 - 2 = 31$$ $$CD = 3x + 9 = 3(11) + 9 = 33 + 9 = 42$$ 5. **Find $m\angle A$ given $m\angle C = 72^\circ$:** In an isosceles trapezoid, the base angles are congruent in pairs. Angles $A$ and $B$ are one pair, and angles $C$ and $D$ are the other. Also, consecutive angles between the bases are supplementary: $$m\angle A + m\angle D = 180^\circ$$ Since $m\angle C = 72^\circ$ and $m\angle C = m\angle D$ (because trapezoid is isosceles), then: $$m\angle D = 72^\circ$$ Therefore: $$m\angle A = 180^\circ - 72^\circ = 108^\circ$$ **Final answers:** $$x = 11$$ $$m\angle A = 108^\circ$$