Question

In the given figure, \(\Delta ABE \cong \Delta ACD\). Prove that \(\Delta ADE \sim \Delta ABC\).

Hint:

Whenever you need to prove similarity inside a larger triangle where parts are congruent, look for a shared angle.
The shared angle \( \angle A \) is almost always the key.
Establishing the proportional relation of the adjacent sides \( AD/AB = AE/AC \) immediately allows the use of the SAS similarity criterion.

Answer 1

Step 1: Understanding the Question:
The topic of this question is Triangles (Congruency and Similarity of Triangles).
We are given that triangle \( ABE \) is congruent to triangle \( ACD \).
We need to prove that the smaller triangle at the top, \( ADE \), is similar to the larger triangle \( ABC \).

Step 2: Key Formula or Approach:
- Since \( \Delta ABE \cong \Delta ACD \), their corresponding parts must be equal by CPCT (Corresponding Parts of Congruent Triangles).
- Specifically, we will use the equalities of the sides: \( AB = AC \) and \( AD = AE \).
- We can then set up ratios of the corresponding sides of \( \Delta ADE \) and \( \Delta ABC \) and use the Side-Angle-Side (SAS) similarity criterion.

Step 3: Detailed Explanation:
1. We are given:
\[ \Delta ABE \cong \Delta ACD \]
2. By CPCT, the corresponding sides of congruent triangles are equal:
- \( AB = AC \) (Equation 1)
- \( AE = AD \implies AD = AE \) (Equation 2)
3. Divide Equation 2 by Equation 1 to find the ratio of the corresponding sides:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]
4. Now, let us compare the triangles \( \Delta ADE \) and \( \Delta ABC \):
- The ratio of the sides containing the shared angle is equal:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]
- The angle \( \angle A \) is common to both triangles:
\[ \angle DAE = \angle BAC \quad \text{(Common angle)} \]
5. By the Side-Angle-Side (SAS) similarity criterion:
\[ \Delta ADE \sim \Delta ABC \]
This completes the formal geometric proof.

Step 4: Final Answer:
Hence, it is proved that \(\Delta ADE \sim \Delta ABC\) by SAS similarity.