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Let's analyze the problem:

We are given a quadrilateral \( CDEF \) that forms a rhombus, and we know the following information:

- \( \angle FCG = 4b + 26^\circ \)
- \( \angle CFE = 60^\circ \)

Since \( CDEF \) is a rhombus, its properties are as follows:
1. All sides of a rhombus are equal.
2. The diagonals of a rhombus bisect each other at right angles (90°).
3. The diagonals of a rhombus bisect the angles of the rhombus.

### Step 1: Use angle properties of the rhombus
The angle \( \angle CFE = 60^\circ \) represents an exterior angle of the triangle \( \triangle CFG \). Since the diagonals bisect the angles in a rhombus, the diagonal \( CF \) bisects \( \angle C \), and we know:

\[
\angle FCG = \angle GCE = 4b + 26^\circ
\]

The angles in triangle \( \triangle CFG \) must sum up to 180°:

\[
\angle FCG + \angle CFG + \angle GCF = 180^\circ
\]

Since \( \angle CFG = 90^\circ \) (because the diagonals of a rhombus are perpendicular), we can write:

\[
(4b + 26^\circ) + 90^\circ + (4b + 26^\circ) = 180^\circ
\]

### Step 2: Solve the equation
Simplifying this:

\[
2(4b + 26^\circ) + 90^\circ = 180^\circ
\]
\[
8b + 52^\circ + 90^\circ = 180^\circ
\]
\[
8b + 142^\circ = 180^\circ
\]
\[
8b = 180^\circ - 142^\circ
\]
\[
8b = 38^\circ
\]
\[
b = \frac{38^\circ}{8} = 4.75^\circ
\]

Thus, the value of \( b \) is \( 4.75^\circ \).

Let me know if you need further details or have any questions.

Here are 5 related questions:
1. What are the properties of diagonals in a rhombus?
2. How do you find the angles of a triangle when given two angles?
3. What is the sum of the interior angles of a triangle?
4. How are exterior angles related to the interior angles of polygons?
5. Can you derive the length of the diagonals using trigonometry if all sides of the rhombus are known?

**Tip:** In a rhombus, remember that the diagonals always bisect the interior angles, and they intersect at 90°.

Quadrilateral CDEF is a rhombus and m∠FCG = 4b + 26°. What is the value of b?

Learn how to solve a problem involving rhombus CDEF and find the value of b, where m∠FCG = 4b + 26° and ∠CFE = 60°. This geometry problem explores the properties of rhombuses and the sum of angles in a triangle. Suitable for students in Grades 8-10.

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