1. **Stating the problem:** We are given two pairs of angles formed by two parallel lines and a transversal. We need to classify the angles, find the value of $x$, and calculate the measures of the angles. 2. **First pair of angles:** - Angle 1: $(6x + 2)^\circ$ - Angle 2: $(11x - 9)^\circ$ Since these angles are corresponding angles formed by parallel lines and a transversal, they are equal: $$6x + 2 = 11x - 9$$ 3. **Solving for $x$:** Move terms to isolate $x$: $$6x + 2 = 11x - 9$$ $$2 + 9 = 11x - 6x$$ $$11 = 5x$$ Divide both sides by 5: $$x = \frac{11}{5} = 2.2$$ 4. **Calculate the measures of the angles:** Angle 1: $$6x + 2 = 6(2.2) + 2 = 13.2 + 2 = 15.2^\circ$$ Angle 2: $$11x - 9 = 11(2.2) - 9 = 24.2 - 9 = 15.2^\circ$$ 5. **Angle classification:** Corresponding angles are equal when lines are parallel. --- 6. **Second pair of angles:** - Angle 1: $(9x + 14)^\circ$ - Angle 2: $(3x + 10)^\circ$ These angles are on the same side of the transversal and inside the parallel lines, so they are consecutive interior angles, which are supplementary: $$ (9x + 14) + (3x + 10) = 180 $$ 7. **Solving for $x$:** $$9x + 14 + 3x + 10 = 180$$ $$12x + 24 = 180$$ $$12x = 180 - 24$$ $$12x = 156$$ Divide both sides by 12: $$x = \frac{156}{12} = 13$$ 8. **Calculate the measure of Angle 1:** $$9x + 14 = 9(13) + 14 = 117 + 14 = 131^\circ$$ 9. **Angle classification:** Consecutive interior angles are supplementary. **Final answers:** - First pair: - Angle Classification: Corresponding - Value of $x$: 2.2 - Measure of Angle 1: $15.2^\circ$ - Measure of Angle 2: $15.2^\circ$ - Second pair: - Angle Classification: Consecutive interior - Value of $x$: 13 - Measure of Angle 1: $131^\circ$