BD bisects angle ABC. Find the measure of ABD. Angle ABD= (3x-17) and Angle CBD = (7x-39)

Given that \( BD \) bisects \( \angle ABC \), the measures of \( \angle ABD \) and \( \angle CBD \) are equal.

So, we can set up the equation:
\[
\angle ABD = \angle CBD
\]

Given:
\[
\angle ABD = 3x - 17
\]
\[
\angle CBD = 7x - 39
\]

Since \( BD \) bisects \( \angle ABC \), we equate the two expressions:
\[
3x - 17 = 7x - 39
\]

Now, solve for \( x \):

1. Subtract \( 3x \) from both sides:
\[
-17 = 4x - 39
\]

2. Add 39 to both sides:
\[
22 = 4x
\]

3. Divide both sides by 4:
\[
x = \frac{22}{4} = 5.5
\]

Now, substitute \( x = 5.5 \) back into the expression for \( \angle ABD \):

\[
\angle ABD = 3(5.5) - 17 = 16.5 - 17 = -0.5
\]

Given that this value is not physically meaningful for an angle, recheck the setup. If the expressions were correctly assigned, the procedure above would be valid. Consider that angles are usually positive in standard interpretation. Without specific contextual reasons, a negative angle might be an error in formulation. 

Would you like to clarify any of these steps or understand an alternative approach? Here are some questions you might find useful:

1. Could we re-evaluate the setup of the angle equations?
2. Do you need further explanation on angle bisectors?
3. Would you like to explore another similar problem?
4. Could there be additional constraints on \( x \) we haven't considered?
5. Should we discuss potential negative angle interpretations?

**Tip**: When solving for angles, always check if the context implies positive values unless specified otherwise.

Explore how to find the measures of angles ABD and CBD using the Angle Bisector Theorem. Step-by-step solution provided with equations setup and solving for the variable x. Suitable for Grades 9-10.

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