1. **State the problem:** We are given two triangles with some angles expressed in terms of $x$ and some known angles. We need to write equations, solve for $x$, and find the missing angle measures. --- ### First triangle: Given angles: - $m\angle U = 80^\circ$ (Acute triangle) - $m\angle B = 44^\circ$ (Right triangle, so one angle is $90^\circ$) - Missing angle $m\angle C = 7x - 10$ Since the triangle is right, one angle is $90^\circ$. The sum of angles in any triangle is $180^\circ$. **Equation:** $$7x - 10 + 3x + 90 = 180$$ Simplify: $$10x - 10 + 90 = 180$$ $$10x + 80 = 180$$ Subtract 80 from both sides: $$10x + \cancel{80} - \cancel{80} = 180 - 80$$ $$10x = 100$$ Divide both sides by 10: $$\frac{10x}{\cancel{10}} = \frac{100}{\cancel{10}}$$ $$x = 10$$ Find missing angles: $$m\angle C = 7x - 10 = 7(10) - 10 = 70 - 10 = 60^\circ$$ $$m\angle A = 3x = 3(10) = 30^\circ$$ Check sum: $$80 + 30 + 70 = 180^\circ$$ (matches) --- ### Second triangle: Given: - $m\angle Q = 83^\circ$ (Obtuse triangle) - $m\angle T = 73^\circ$ - Missing angle $m\angle A = 6x + 3$ Sum of angles: $$6x + 3 + 83 + 73 = 180$$ Simplify: $$6x + 3 + 156 = 180$$ $$6x + 159 = 180$$ Subtract 159 from both sides: $$6x + \cancel{159} - \cancel{159} = 180 - 159$$ $$6x = 21$$ Divide both sides by 6: $$\frac{6x}{\cancel{6}} = \frac{21}{\cancel{6}}$$ $$x = 3.5$$ Find missing angle: $$m\angle A = 6x + 3 = 6(3.5) + 3 = 21 + 3 = 24^\circ$$ Check sum: $$83 + 73 + 24 = 180^\circ$$ (matches) --- **Final answers:** - For the first triangle: $x = 10$, $m\angle C = 60^\circ$, $m\angle A = 30^\circ$ - For the second triangle: $x = 3.5$, $m\angle A = 24^\circ$, $m\angle Q = 83^\circ$, $m\angle T = 73^\circ$