Question

what is ( mangle def )?
( mangle def=square^{circ} )

Explanation:

Step1: Use Hypotenuse - Leg (HL) Congruence

Since \(GF = GD = 27\), \(GE=GE\) (common side), \(\angle GFE=\angle GDE = 90^{\circ}\), by HL (Hypotenuse - Leg) congruence, \(\triangle GFE\cong\triangle GDE\).

Step2: Use Congruent Angles Property

If \(\triangle GFE\cong\triangle GDE\), then \(\angle GEF=\angle GED\).

Step3: Calculate \(\angle DEF\)

Let \(\angle GEF = x\) and \(\angle GED=x\). We know that \(\angle FED=\angle FEG+\angle GED\). Given \(\angle FEG = 56^{\circ}\), then \(\angle DEF=2\times56^{\circ}\)

Answer:

\(112\)