1. **State the problem:** We are given two parallel lines $b$ and $c$ cut by a transversal $a$. We know the measures of two angles expressed in terms of $x$: the upper-left angle at the top intersection is $(13x+9)^\circ$ and the lower-left angle at the bottom intersection is $(5x+9)^\circ$. We need to find the measure of angle 6. 2. **Identify the relationship:** Since lines $b$ and $c$ are parallel and cut by transversal $a$, corresponding angles are congruent. The upper-left angle at the top intersection corresponds to the lower-left angle at the bottom intersection. 3. **Set up the equation:** $$ 13x + 9 = 5x + 9 $$ 4. **Solve for $x$:** $$ 13x + 9 = 5x + 9 \\ 13x - 5x = 9 - 9 \\ 8x = 0 \\ x = \frac{0}{8} = 0 $$ 5. **Find the measure of angle 6:** Angle 6 is vertically opposite to angle $(5x+9)^\circ$ at the bottom intersection, so they are equal. Substitute $x=0$: $$ 5(0) + 9 = 9^\circ $$ 6. **Check the options:** The measure of angle 6 is $9^\circ$, which is not among the given options. This suggests a different angle relationship must be used. 7. **Re-examine the problem:** Angles $(13x+9)^\circ$ and $(5x+9)^\circ$ are alternate interior angles, so they are equal. Set equation: $$ 13x + 9 = 180 - (5x + 9) $$ Because they are on opposite sides of the transversal but inside the parallel lines, alternate interior angles are equal, so this is incorrect. Instead, they are supplementary if they are consecutive interior angles. If they are consecutive interior angles, then: $$ (13x + 9) + (5x + 9) = 180 $$ Simplify: $$ 18x + 18 = 180 \\ 18x = 162 \\ x = 9 $$ 8. **Calculate angle 6:** $$ 5x + 9 = 5(9) + 9 = 45 + 9 = 54^\circ $$ Angle 6 is vertically opposite to this angle, so: $$ m\angle 6 = 54^\circ $$ **Final answer:** $m\angle 6 = 54^\circ$