1. **Problem Statement:** Determine the values of the missing variables and the measures of each unknown angle for the given angle relationships involving parallel lines, transversals, triangles, and polygons. 2. **Key Angle Theorems and Formulas:** - Corresponding angles are equal when two parallel lines are cut by a transversal. - Alternate interior angles are equal. - Consecutive interior angles are supplementary (sum to 180°). - The sum of angles in a triangle is 180°. - Opposite angles in a parallelogram are equal. 3. **Solve the first problem (a):** Given: $3x + x + 2 = 5x - 6 = 90$ This seems to be a miswritten equation. The correct interpretation is likely $3x + x + 2 = 90$ and $5x - 6 = 90$. Step 1: Solve $3x + x + 2 = 90$ $$4x + 2 = 90$$ $$4x = 90 - 2$$ $$4x = 88$$ $$x = \frac{88}{4} = 22$$ Step 2: Solve $5x - 6 = 90$ $$5x = 90 + 6$$ $$5x = 96$$ $$x = \frac{96}{5} = 19.2$$ Since $x$ cannot have two different values, check which equation applies to the angle measure. 4. **Solve problem (b):** Angles on opposite sides of the transversal are equal: $$10x - 13 = 4x + 10$$ $$10x - 4x = 10 + 13$$ $$6x = 23$$ $$x = \frac{23}{6} \approx 3.83$$ 5. **Solve problem (c):** Alternate interior angles equal: $$15x - 200 = 5x$$ $$15x - 5x = 200$$ $$10x = 200$$ $$x = 20$$ 6. **Solve problem (d):** Right triangle sides: $x + 8$ and $x + 5$ (likely legs or angles). If these are angles, sum with right angle 90°: $$x + 8 + x + 5 + 90 = 180$$ $$2x + 13 = 90$$ $$2x = 77$$ $$x = 38.5$$ 7. **Solve problem (e):** Angles on opposite sides of transversal are equal: $$3x - 10 = 2x - 24$$ $$3x - 2x = -24 + 10$$ $$x = -14$$ Negative $x$ is not possible for angle measures, recheck problem context. 8. **Solve problem (f):** Given $x = 11$, find $3x - 14$: $$3(11) - 14 = 33 - 14 = 19$$ 9. **Solve problem (g):** Parallelogram opposite sides equal: $$2x + 3 = x + 18$$ $$2x - x = 18 - 3$$ $$x = 15$$ 10. **Solve problem (h):** Triangle angles sum to 180°: $$6x + 14 + 5x + 12 + \text{third angle} = 180$$ Sum known angles: $$11x + 26 + \text{third angle} = 180$$ Without third angle, cannot solve for $x$. 11. **Solve problem (i):** Alternate interior angles equal: $$8x = 6x + 1$$ $$8x - 6x = 1$$ $$2x = 1$$ $$x = 0.5$$ 12. **Solve problem (j):** Rectangle or parallelogram adjacent angles supplementary: $$x + 7 + x + 2 = 180$$ $$2x + 9 = 180$$ $$2x = 171$$ $$x = 85.5$$ 13. **Solve problem (k):** Alternate interior angles equal: $$8x - 17 = 6x + 9$$ $$8x - 6x = 9 + 17$$ $$2x = 26$$ $$x = 13$$ **Final answers:** - a) $x = 22$ or $x = 19.2$ (check context) - b) $x \approx 3.83$ - c) $x = 20$ - d) $x = 38.5$ - e) $x = -14$ (check problem) - f) $x = 11$, angle $= 19$ - g) $x = 15$ - h) Insufficient data - i) $x = 0.5$ - j) $x = 85.5$ - k) $x = 13$