1. **State the problem:** We are given a triangle with vertices A, B, and C. The angle at vertex A is given as $2x - 20$ degrees. The sides opposite vertices B and C are labeled with $x$. We need to find the measures of angles $\angle X$ and $\angle A$, and determine the type of triangle. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always 180 degrees. 3. **Set up the equation:** Let the angles at vertices B and C be $x$ each (since sides opposite B and C are labeled $x$, we assume the triangle is isosceles with angles at B and C equal). Then, $$\angle A + \angle B + \angle C = 180$$ $$ (2x - 20) + x + x = 180$$ 4. **Simplify the equation:** $$2x - 20 + x + x = 180$$ $$4x - 20 = 180$$ 5. **Solve for $x$:** Add 20 to both sides: $$4x - \cancel{20} + 20 = 180 + 20$$ $$4x = 200$$ Divide both sides by 4: $$\frac{\cancel{4}x}{\cancel{4}} = \frac{200}{4}$$ $$x = 50$$ 6. **Find the angles:** $$\angle A = 2x - 20 = 2(50) - 20 = 100 - 20 = 80$$ $$\angle B = x = 50$$ $$\angle C = x = 50$$ 7. **Determine the type of triangle:** Since two angles are equal ($50^\circ$ and $50^\circ$), the triangle is isosceles. **Final answers:** $$m\angle X = 50^\circ, \quad m\angle A = 80^\circ$$ The triangle is isosceles because it has two equal angles.