22. Identify the congruence criteria for triangles MEF and NFE based on the markings. Provide a step-by-step explanation.

Step-by-step explanation:

• Step 1: Analyze the markings. In triangle MEF, we see that angle M and angle N are marked with a single arc, indicating $$\angle M = \angle N$$. Side EF is common to both triangles MEF and NFE. Angle MEF and angle NFE are marked with double arcs, indicating $$\angle MEF = \angle NFE$$.
• Step 2: Identify the congruence criterion. We have two angles and a side. The common side EF is opposite to angle M in triangle MEF and opposite to angle N in triangle NFE. Since $$\angle M = \angle N$$, these are opposite the common side. The other pair of equal angles is $$\angle MEF$$ and $$\angle NFE$$.
• Step 3: Apply the ASA congruence criterion. We have angle $$\angle M = \angle N$$ (Angle), side EF = EF (Side), and angle $$\angle MEF = \angle NFE$$ (Angle). The side EF is included between $$\angle MEF$$ and the angle at M (which is not directly given as equal). However, if we consider the angles adjacent to the common side EF, we have $$\angle MEF$$ and $$\angle NFE$$. We also have $$\angle M$$ and $$\angle N$$.
• Step 4: Consider the AAS congruence criterion. We have $$\angle M = \angle N$$ (Angle), $$\angle MEF = \angle NFE$$ (Angle), and EF = EF (Side). The side EF is not included between the two angles $$\angle M$$ and $$\angle MEF$$, nor between $$\angle N$$ and $$\angle NFE$$. However, EF is opposite to $$\angle M$$ in $$\triangle MEF$$ and opposite to $$\angle N$$ in $$\triangle NFE$$.
• Step 5: Re-examine angles. In $$\triangle MEF$$, we have $$\angle M$$, $$\angle MEF$$, and $$\angle MFE$$. In $$\triangle NFE$$, we have $$\angle N$$, $$\angle NFE$$, and $$\angle NEF$$. We are given $$\angle M = \angle N$$ and $$\angle MEF = \angle NFE$$. The common side is EF.
• Step 6: Apply AAS. In $$\triangle MEF$$, we have angle $$\angle M$$, angle $$\angle MEF$$, and side EF. In $$\triangle NFE$$, we have angle $$\angle N$$, angle $$\angle NFE$$, and side EF. Since $$\angle M = \angle N$$ and $$\angle MEF = \angle NFE$$, and the side EF is not between the angles, we can try AAS. In $$\triangle MEF$$, we have $$\angle N$$ and $$\angle NFE$$. The side opposite $$\angle N$$ is EF. The side opposite $$\angle NFE$$ is ME.
• Step 7: Correct application of AAS. In $$\triangle MEF$$: Angle $$\angle M$$, Angle $$\angle MEF$$, Side EF. In $$\triangle NFE$$: Angle $$\angle N$$, Angle $$\angle NFE$$, Side EF. Given $$\angle M = \angle N$$ and $$\angle MEF = \angle NFE$$. Side EF is common. Consider $$\triangle MEF$$. We have angle $$\angle M$$, angle $$\angle MEF$$. Side EF is given. Consider $$\triangle NFE$$. We have angle $$\angle N$$, angle $$\angle NFE$$. Side EF is given. Since $$\angle M = \angle N$$ and $$\angle MEF = \angle NFE$$, and side EF is common. We can use the AAS (Angle-Angle-Side) congruence criterion if EF is opposite to one of the angles. EF is opposite to $$\angle M$$ in $$\triangle MEF$$ and opposite to $$\angle N$$ in $$\triangle NFE$$. Since $$\angle M = \angle N$$, and we have another pair of equal angles $$\angle MEF = \angle NFE$$, and the side EF is opposite to $$\angle M$$ (and $$\angle N$$), the triangles are congruent by AAS.

Answer: Triangles MEF and NFE are congruent by the AAS (Angle-Angle-Side) congruence criterion.