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Problem 87

# Given: $$\angle A \cong \angle D, A E \cong D E, A C \cong \underline{D B} .$$ Prove: $$\Delta A B E \cong \triangle D C E$$.

Expert verified
Given: $$\angle A \cong \angle D$$, $$A E \cong D E$$, and $$A C \cong D B$$. Applying ASA Postulate since: 1. $$\angle A \cong \angle D$$ - Angle 2. $$A E \cong D E$$ - Side (included between the angles) 3. $$\angle A B E \cong \angle D C E$$ - Angle (Vertical angles) Hence, by ASA Congruence Postulate, $$\Delta A B E \cong \triangle D C E$$.
See the step by step solution

## Step 1: Identify the two triangles

We are given two triangles, $$\Delta A B E$$ and $$\Delta D C E$$. Our goal is to prove that they are congruent.

## Step 2: List the given information

We are given the following: 1. $$\angle A \cong \angle D$$ 2. $$A E \cong D E$$ 3. $$A C \cong \underline{D B}$$

## Step 3: Use the Angle-Side-Angle Postulate

In order to prove the congruence between the triangles, we'll use the Angle-Side-Angle (ASA) Postulate. According to this postulate, if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. Now let's analyze the given information according to ASA postulate: 1. $$\angle A \cong \angle D$$ - Angle 2. $$A E \cong D E$$ - Side (included between the angles) 3. From point 3, we have $$A C \cong D B$$, and B and D are corresponding vertices in $$\Delta A B E$$ and $$\Delta D C E$$ respectively, therefore $$\angle A B E \cong \angle D C E$$ (Since vertical angles are congruent) - Angle We have Angle-Side-Angle, so by the ASA Postulate, we can conclude:

## Step 4: Write the final conclusion

Since $$\angle A \cong \angle D$$, $$A E \cong D E$$, and $$\angle A B E \cong \angle D C E$$, by the ASA Congruence Postulate, we can conclude that $$\Delta A B E \cong \triangle D C E$$.

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