Let’s solve the given problem step by step.

Problem Analysis:

We are given:

1. ∠CDE is a straight angle, meaning $m∠CDE = 180^\circ$.
2. $\overline{DE}$ bisects ∠GDH, so $m∠GDE = m∠EDH$.
3. $m∠GDE = (8x - 1)^\circ$, $m∠EDH = (6x + 15)^\circ$, and $m∠CDF = 43^\circ$.

We need to:

1. Find $x$.
2. Calculate $m∠GDH$, $m∠FDH$, and $m∠FDE$.

Step 1: Solve for $x$

Since $\overline{DE}$ bisects $∠GDH$, we know: $m∠GDE = m∠EDH.$ Substitute the given expressions: $8x - 1 = 6x + 15.$ Solve for $x$: $8x - 6x = 15 + 1,$ $2x = 16,$ $x = 8.$

Step 2: Find $m∠GDE$ and $m∠EDH$

Substitute $x = 8$ into the expressions for $m∠GDE$ and $m∠EDH$: $m∠GDE = 8(8) - 1 = 64 - 1 = 63^\circ,$ $m∠EDH = 6(8) + 15 = 48 + 15 = 63^\circ.$

Thus, $m∠GDH = m∠GDE + m∠EDH = 63^\circ + 63^\circ = 126^\circ$.

Step 3: Find $m∠FDH$

We know: $m∠CDE = m∠CDF + m∠FDH.$ Substitute the known values: $180 = 43 + m∠FDH.$ Solve for $m∠FDH$: $m∠FDH = 180 - 43 = 137^\circ.$

Step 4: Find $m∠FDE$

From the figure: $m∠FDE = m∠FDH - m∠EDH.$ Substitute the values: $m∠FDE = 137 - 63 = 74^\circ.$

Final Answers:

1. $x = 8$,
2. $m∠GDH = 126^\circ$,
3. $m∠FDH = 137^\circ$,
4. $m∠FDE = 74^\circ$.

Would you like further explanation or details?

Related Questions:

1. How do you determine if two angles are supplementary or complementary?
2. What are the properties of angle bisectors in geometry?
3. How do you calculate an angle when given a straight angle and one of its parts?
4. What is the significance of substituting known values into angle equations?
5. How can you verify the solution geometrically or algebraically?

Tip:

Always re-check calculations by substituting your final results into the original conditions to ensure accuracy.