In Fig. 6.37, if ∆ ABE ≅ ∆ ACD, show that ∆ ADE ~ ∆ ABC.

We are tasked with proving that ∆ADE is similar to ∆ABC, given that ∆ABE ≅ ∆ACD. Let's solve step by step, referring to the figure (in fig 6.37) to make the solution easy to understand.

Step 1: Use the given information

In fig 6.37, it is given that ∆ABE ≅ ∆ACD. From the properties of congruent triangles, we know:

1. AD = AE (corresponding sides of ∆ABE and ∆ACD) ..........(1)
2. AB = AC (corresponding sides of ∆ABE and ∆ACD) ..........(2)

Step 2: Compare ∆ADE and ∆ABC

Now, consider the two triangles ∆ADE and ∆ABC from fig 6.37. To prove their similarity, we need to show that they satisfy the SAS similarity criterion.

Step 2.1: Ratios of corresponding sides

From equations (1) and (2):

AD / AB = AE / AC

This establishes that the corresponding sides of ∆ADE and ∆ABC are proportional.

Step 2.2: Common angle

In fig 6.37, the angle ∠DAE is the same as the angle ∠BAC because they are vertically opposite angles.

Step 3: Apply SAS similarity criterion

Using the proportionality of sides (from Step 2.1) and the common angle ∠DAE = ∠BAC (from Step 2.2), we can conclude that:

∆ADE ~ ∆ABC (by SAS similarity criterion).

Final Answer:

Thus, it is proven that ∆ADE ~ ∆ABC based on the information provided in fig 6.37.