Applying the SAS congruence theorem and the definition of angle bisector, the complete proof is shown in the image attached below.
What is the SAS Theorem?
The SAS theorem (side-angle-side theorem) state that two triangles are congruent to each other if they have two pairs of congruent sides that are corresponding to each other and a pair of congruent angles that are corresponding to each other.
What is an Angle Bisector?
A line segment that divides an angle into two equal parts is referred to as an angle bisector.
From the proof given, we know that;
m∠DEK = 2(m∠DEF)
m∠DEK = 2(44) = 88°
With the given information and using the reflexive property, we know that triangles DEF and KEF have two pairs of congruent sides that are corresponding to each other and a pair of congruent angles that are corresponding to each other.
Therefore, ΔDEF ≅ ΔKEF based on the SAS congruence theorem. Because they are congruent triangles, DF = KF based on the CPCTC theorem.
Then, KF = 1/2(DK) = 1/2(16) = 8.
The complete proof is show in the image attached below.
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