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To solve the problem of finding \( m\angle ABD \) and \( m\angle CBD \), let's use the information provided:

1. \( m\angle ABC = 77^\circ \)
2. \( m\angle ABD = (3x + 22)^\circ \)
3. \( m\angle CBD = (5x - 17)^\circ \)

From the diagram, we know that:

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

Substituting the given expressions:

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

Now, let's solve this equation:

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

Simplify the equation:

\[
77 = 8x + 5
\]

Subtract 5 from both sides:

\[
72 = 8x
\]

Now, divide by 8:

\[
x = 9
\]

Now that we have \( x = 9 \), we can find the measures of the angles:

1. For \( m\angle ABD \):

\[
m\angle ABD = 3x + 22 = 3(9) + 22 = 27 + 22 = 49^\circ
\]

2. For \( m\angle CBD \):

\[
m\angle CBD = 5x - 17 = 5(9) - 17 = 45 - 17 = 28^\circ
\]

So, the measures are:

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

Would you like any further details or explanations? 

Here are some related questions for further practice:

1. What would happen to the measures of \( m\angle ABD \) and \( m\angle CBD \) if \( m\angle ABC \) was increased by 10 degrees?
2. If \( m\angle ABC = 77^\circ \), what are the individual measures of \( m\angle ABD \) and \( m\angle CBD \) if \( 2x + 20^\circ \) and \( 6x - 23^\circ \) are used as expressions for the angles?
3. Can you find \( m\angle ABC \) if \( m\angle ABD = 4x + 15 \) and \( m\angle CBD = 2x - 5 \), and \( x = 10 \)?
4. What would the expressions for \( m\angle ABD \) and \( m\angle CBD \) be if the total angle \( m\angle ABC \) was divided in a 2:3 ratio?
5. How would the values of \( x \) change if \( m\angle ABD \) was given by \( 2x + 40 \) and \( m\angle CBD \) by \( 3x - 10 \)?

**Tip:** When working with angles in diagrams, always check that the sum of the smaller angles matches the larger angle when summed.

Find m∠ABD and m∠CBD given that m∠ABC = 77°, m∠ABD = (3x + 22)° and m∠CBD = (5x - 17)°.

Learn how to solve for angles m∠ABD and m∠CBD using algebraic expressions, given m∠ABC = 77°. This step-by-step solution covers angle addition and basic algebra. Suitable for students in Grades 8-10.

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