Question

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in the figure below, ∠cfd ≅ ∠edf.
to prove δcdf ≅ δefd by the aas congruence criteria, using pairs of congruent angles and the reflexive property, it must also be known that angles are congruent.
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Explanation:

Step1: Recall AAS Criterion

AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the two triangles are congruent.

We know that $\angle CFD\cong\angle EDF$ (given). The side $DF$ is common to both $\triangle CDF$ and $\triangle EFD$, so by the reflexive property, $DF\cong DF$.

For AAS, we need another pair of angles. Let's consider the angles in the triangles. In $\triangle CDF$ and $\triangle EFD$, we already have one pair of angles ($\angle CFD$ and $\angle EDF$) and the side $DF$. To apply AAS, we need to have another pair of angles. The angles $\angle C$ and $\angle E$ should be congruent. Because in $\triangle CDF$, the angles are $\angle C$, $\angle CFD$, and $\angle CDF$; in $\triangle EFD$, the angles are $\angle E$, $\angle EDF$, and $\angle EFD$. We know $\angle CFD\cong\angle EDF$ and $DF$ is common. If $\angle C\cong\angle E$, then with the side $DF$ (non - included between the two angles), we can apply AAS.

Step2: Analyze the Triangles

Looking at the figure, $\triangle CDF$ and $\triangle EFD$ share the side $DF$. We have $\angle CFD\cong\angle EDF$. For AAS, we need two angles and a non - included side. The non - included side here is $DF$. So we need another pair of angles. The angles $\angle C$ (in $\triangle CDF$) and $\angle E$ (in $\triangle EFD$) must be congruent.

Answer:

$\angle C$ and $\angle E$