Math Snap

STEP 1

1. $C$ is the midpoint of $\overline{A E}$ and $\overline{B D}$.
2. We need to prove $\triangle A B C \cong \triangle E D C$.

STEP 2

1. Use the definition of midpoint.
2. Establish congruent segments.
3. Use the Vertical Angles Congruence Theorem.
4. Apply the SAS Congruence Theorem.

STEP 3

Since $C$ is the midpoint of $\overline{A E}$ and $\overline{B D}$, by definition of midpoint, we have:
$$\overline{A C} \cong \overline{C E} \quad \text{and} \quad \overline{B C} \cong \overline{C D}$$

STEP 4

Using the congruent segments from the midpoint definition, we can establish:
$$\overline{A C} \cong \overline{C E} \quad \text{and} \quad \overline{B C} \cong \overline{C D}$$

STEP 5

By the Vertical Angles Congruence Theorem, the angles $\angle A C B$ and $\angle E C D$ are congruent because they are vertical angles:
$$\angle A C B \cong \angle E C D$$

Now, we apply the SAS Congruence Theorem. We have two sides and the included angle congruent:
1. $\overline{A C} \cong \overline{C E}$
2. $\angle A C B \cong \angle E C D$
3. $\overline{B C} \cong \overline{C D}$
Thus, by SAS, $\triangle A B C \cong \triangle E D C$.
The triangles are congruent:
$$\triangle A B C \cong \triangle E D C$$