AC is a line. line BE bisect angle ABD. angle ABE=4x+7 and angle EBD=6x-17. Find the X, mangleABE, mangleEBD and mangleDBC

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.

AC is a line. Line BE bisects angle ABD. Angle ABE = 4x + 7 and angle EBD = 6x - 17. Find the value of x, m∠ABE, m∠EBD, and m∠DBC.

Learn how to solve for the value of x, m∠ABE, m∠EBD, and m∠DBC in a geometric problem involving an angle bisector. This problem includes key algebraic and geometric concepts such as the Angle Bisector Theorem and linear pairs. Suitable for students in Grades 8-10.

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