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The problem asks to find the measure of \( \angle ADC \) in the rhombus \( ABCD \).

Given:
- \( ABCD \) is a rhombus.
- Angles and expressions for different parts of the rhombus are labeled in the diagram.

### Key properties of a rhombus:
1. **Opposite angles are equal.**
2. **Diagonals bisect each other at right angles (90°).**
3. **Each diagonal bisects the angles at the vertices.**

#### Analysis of the Diagram:
- \( \angle ABC = 12^\circ \)
- \( \angle BCF = 4y - 1^\circ \)
- \( \angle ACD = 7x + 2^\circ \)
- \( \angle ADC = 4x + 17^\circ \)

### Step-by-step process:

#### Step 1: Use the property that diagonals bisect the angles.
Since \( ABCD \) is a rhombus, the diagonals bisect each other at right angles, meaning that each angle formed at \( D \), such as \( \angle ADC \), should fit into the angle-sum rule.

We know that the diagonals split the rhombus into four right-angled triangles. Therefore, using properties of the rhombus and solving the equation for the unknowns \( x \) and \( y \) can give us more details.

Let me set up the expressions for the angles and solve for \( \angle ADC \).After solving the equations, we find that \( \angle ADC \) is approximately:

\[
\angle ADC = \frac{471}{11} \approx 42.82^\circ
\]

Would you like more details on the steps? Here are five related questions to explore:

1. How can we verify the calculated value of \( \angle ADC \) using properties of rhombus?
2. What are the properties of the diagonals in a rhombus and how do they affect angles?
3. How can other angles of the rhombus be calculated from this solution?
4. How does knowing that diagonals bisect the angles help in solving for unknowns?
5. What role do supplementary angles play in a rhombus?

**Tip:** Always use the fact that diagonals of a rhombus are perpendicular and bisect the vertex angles when solving for unknown angles.

ABCD is a rhombus. Find the measure of \( \angle ADC \).

Learn how to find the measure of \( \angle ADC \) in a rhombus using algebraic expressions and properties of rhombus diagonals. Suitable for students in Grades 9-11, this problem integrates key geometry concepts.

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