Question

find the missing measurement of the rhombus.

\\(60^\circ\\)
\\(45^\circ\\)
\\(120^\circ\\)
\\(30^\circ\\)

Explanation:

Identify the given properties of the shape

Using the Parallelogram Properties knowledge point

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Analyze triangle FGE

Since \(FG = FE\), \(\triangle FGE\) is an isosceles triangle.

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Determine the missing angle

The question asks for the angle marked with a "?" which is \(\angle DEG\).
Using the Parallelogram Properties knowledge point

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Alternatively, since diagonal \(GE\) bisects the interior angles of the rhombus:

$$ \angle DEG = \frac{1}{2}\angle FED = \frac{1}{2}(120^\circ) = 30^\circ $$

Let's re-evaluate the angle bisector property of a rhombus:
The diagonals of a rhombus bisect its angles.
The angle \(\angle FED\) is opposite to \(\angle FGD\).
Since \(\angle F = 60^\circ\), the consecutive angle \(\angle FED = 180^\circ - 60^\circ = 120^\circ\).
The diagonal \(GE\) bisects \(\angle FED\).
Therefore, the angle marked with "?" which is \(\angle DEG\) is:

$$ \angle DEG = \frac{120^\circ}{2} = 60^\circ $$

Wait, let's verify:
In \(\triangle FGE\), \(FG = FE\), so it is isosceles. Since \(\angle F = 60^\circ\), \(\triangle FGE\) is equilateral, meaning \(\angle FEG = 60^\circ\).
Since diagonal \(GE\) bisects \(\angle FED\), we have \(\angle FEG = \angle DEG = 60^\circ\).
Thus, the missing angle is \(60^\circ\).

Answer:

• (A) \(60^\circ\) (Correct answer)
• (B) \(45^\circ\)
• (C) \(120^\circ\)
• (D) \(30^\circ\)