We are given that line BE bisects ∠ABD, and we know the measures of ∠ABE and ∠EBD in terms of $x$. Here's how we solve this step by step:

Step 1: Use the angle bisector information

Since BE bisects ∠ABD, we know that ∠ABE = ∠EBD. Therefore, we can set the given expressions for these angles equal to each other: $4x + 7 = 6x - 17$

Step 2: Solve for $x$

We solve the equation by first subtracting $4x$ from both sides: $7 = 2x - 17$

Next, add 17 to both sides: $24 = 2x$

Now, divide both sides by 2: $x = 12$

Step 3: Calculate ∠ABE and ∠EBD

Now that we know $x = 12$, we can substitute this value into the expressions for ∠ABE and ∠EBD.

$\text{m∠ABE} = 4x + 7 = 4(12) + 7 = 48 + 7 = 55^\circ$ $\text{m∠EBD} = 6x - 17 = 6(12) - 17 = 72 - 17 = 55^\circ$

Since ∠ABE = ∠EBD by the bisector, the calculations check out.

Step 4: Calculate ∠ABD

Since BE bisects ∠ABD, the whole angle ∠ABD is: $\text{m∠ABD} = 2 \times \text{m∠ABE} = 2 \times 55^\circ = 110^\circ$

Step 5: Calculate ∠DBC

Since ∠ABD is part of the line AC, and a straight line measures $180^\circ$, we can find ∠DBC by subtracting ∠ABD from 180°: $\text{m∠DBC} = 180^\circ - 110^\circ = 70^\circ$

Final Answer:

• $x = 12$
• $\text{m∠ABE} = 55^\circ$
• $\text{m∠EBD} = 55^\circ$
• $\text{m∠DBC} = 70^\circ$

Would you like more details on any of these steps?

Related Questions:

1. What is the significance of an angle bisector?
2. How does the sum of angles on a straight line relate to solving this problem?
3. How can angle bisectors be applied in triangle geometry?
4. Can the angle sum property of triangles be useful here?
5. What are some common angle relationships in parallel line and transversal problems?

Tip:

When angles are bisected, the two resulting angles are equal, and this property can simplify many geometric problems.