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The problem provides the following information:

- The measure of angle \( \angle ABC = 77^\circ \).
- \( \angle ABD = (3x + 22)^\circ \).
- \( \angle CBD = (5x - 17)^\circ \).

To solve for the measures of \( \angle ABD \) and \( \angle CBD \), we use the fact that:

\[
\angle ABC = \angle ABD + \angle CBD
\]

This gives us the equation:

\[
(3x + 22) + (5x - 17) = 77
\]

Let's solve for \( x \).We have found that \( x = 9 \).

Now, let's substitute \( x = 9 \) into the expressions for \( \angle ABD \) and \( \angle CBD \):

- \( \angle ABD = 3x + 22 \)
- \( \angle CBD = 5x - 17 \)

I'll calculate the values of both angles next.The measures of the angles are:

- \( m\angle ABD = 49^\circ \)
- \( m\angle CBD = 28^\circ \)

Would you like any further details or explanations on this solution?

Here are five related questions for practice:
1. What is the measure of \( \angle DBC \) if \( \angle ABC = 90^\circ \)?
2. If \( \angle ABD = (4x + 10)^\circ \) and \( \angle CBD = (6x - 5)^\circ \), what is \( x \) when \( \angle ABC = 80^\circ \)?
3. How do you calculate angles in a triangle if two angles are given?
4. What are complementary angles, and how are they used in geometry problems?
5. What are the different ways to solve for unknown angles in geometric figures?

**Tip:** Always check that the sum of all angles in a triangle adds up to 180°. This helps in verifying calculations when dealing with angles.

Find m∠ABD and m∠CBD given m∠ABC = 77°. The expressions for m∠ABD and m∠CBD are (3x + 22)° and (5x - 17)° respectively.

Solve for m∠ABD and m∠CBD given m∠ABC = 77°. The expressions for m∠ABD and m∠CBD are (3x + 22)° and (5x - 17)° respectively. Use algebra and the Angle Addition Theorem to find the correct values of x and the angles.

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