Question

In the figure given below, ∠1 = ∠2 and `(BE)/(BC) = (CD)/(AB)`. Prove that ΔBDE ∼ ΔBAC.

Answer 1

Given:

1. ∠1 = ∠2

2. `(BE)/(BC) = (CD)/(AB)`

To Prove: ΔBDE ∼ ΔBAC

Proof:

Step 1: Use the angle equality

In ΔBCD, we are given that ∠1 = ∠2 which are ∠DBC and ∠BCD.

Since the angles are equal, the sides opposite those angles must also be equal:

BD = CD   ...(Sides opposite to equal angles in a triangle are equal)

Step 2: Substitute into the given ratio

We are given the ratio:

`(BE)/(BC) = (CD)/(AB)`

Substituting CD with BD (from Step 1), we get:

`(BE)/(BC) = (BD)/(AB)`

By rearranging the terms, we can write:

`(BD)/(AB) = (BE)/(BC)`

Step 3: Establish Similarity

Now, let’s look at ΔBDE and ΔBAC:

1. `(BD)/(BA) = (BE)/(BC)`  ...(From our substitution in Step 2)

2. ∠B = ∠B   ...(This is a common angle for both triangles)

Since two pairs of corresponding sides are in the same ratio and their included angle is equal, the triangles are similar.

ΔBDE ∼ ΔBAC by the SAS (Side-Angle-Side) similarity criterion.