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Let's break down the problem step by step to find the measures of \( \angle ADC \) and \( \angle DCB \).

### Information given:
- \( \angle A = 43^\circ \)
- \( \angle ABC = 61^\circ \)
- \( \angle BDC = (3y - 13)^\circ \)
- \( \angle ADC = (3y + 7)^\circ \)
- \( \angle DBC = (4x - 1)^\circ \)
- \( \angle DCB = (2x + 17)^\circ \)

### To find \( m\angle ADC \) and \( m\angle DCB \):
We can proceed by using angle sum properties and solving the system of equations.

#### Step 1: Use the triangle sum property for \( \triangle ADC \)
In any triangle, the sum of interior angles is \( 180^\circ \).
- For \( \triangle ADC \): 
\[
\angle A + \angle ADC + \angle DCB = 180^\circ
\]
\[
43^\circ + (3y + 7)^\circ + (2x + 17)^\circ = 180^\circ
\]
Simplifying:
\[
43 + 3y + 7 + 2x + 17 = 180
\]
\[
67 + 3y + 2x = 180
\]
\[
3y + 2x = 113 \quad \text{(Equation 1)}
\]

#### Step 2: Use the triangle sum property for \( \triangle BDC \)
For \( \triangle BDC \):
\[
\angle BDC + \angle DBC + \angle DCB = 180^\circ
\]
\[
(3y - 13)^\circ + (4x - 1)^\circ + (2x + 17)^\circ = 180^\circ
\]
Simplifying:
\[
3y - 13 + 4x - 1 + 2x + 17 = 180
\]
\[
3y + 6x + 3 = 180
\]
\[
3y + 6x = 177 \quad \text{(Equation 2)}
\]

#### Step 3: Solve the system of equations
Now solve these two equations:
1. \( 3y + 2x = 113 \)
2. \( 3y + 6x = 177 \)

Subtract equation 1 from equation 2:
\[
(3y + 6x) - (3y + 2x) = 177 - 113
\]
\[
4x = 64
\]
\[
x = 16
\]

Substitute \( x = 16 \) into equation 1:
\[
3y + 2(16) = 113
\]
\[
3y + 32 = 113
\]
\[
3y = 81
\]
\[
y = 27
\]

#### Step 4: Calculate the angle measures
Now that we have \( x = 16 \) and \( y = 27 \), we can find the angles:
- \( \angle ADC = 3y + 7 = 3(27) + 7 = 81 + 7 = 88^\circ \)
- \( \angle DCB = 2x + 17 = 2(16) + 17 = 32 + 17 = 49^\circ \)

### Final answer:
- \( m\angle ADC = 88^\circ \)
- \( m\angle DCB = 49^\circ \)

Would you like further clarification on any step, or additional details?

### Related questions:
1. How would the solution change if the triangle was not scalene but isosceles?
2. Can you solve a triangle if only one angle and one side are given?
3. How does the exterior angle theorem apply in this scenario?
4. Can this problem be solved using trigonometric functions?
5. How would you solve this if the angles were not explicitly labeled but only described?

### Tip:
Always check your solutions by verifying that the sum of angles in each triangle equals 180°, ensuring consistency.

What are the measures of \(\angle ADC\) and \(\angle DCB\) in the triangle with angles expressed as algebraic terms?

Learn how to solve for the measures of angles ADC and DCB in a triangle using the angle sum property and algebraic methods. The problem involves solving a system of equations derived from angle relationships. Suitable for high school students in Grades 9-11.

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