NCERT Solutions
Class 10 - Mathematics - Chapter 6: TRIANGLES - Exercise 6.3

Question 6

Given: \(\triangle ABE \cong \triangle ACD\). Points D and E lie on sides \(AB\) and \(AC\) respectively.

To prove: \(\triangle ADE \sim \triangle ABC\).

Step 1: Use congruence to get equal sides

From \(\triangle ABE \cong \triangle ACD\), we have by CPCT (Corresponding Parts of Congruent Triangles):

\[AB = AC, \quad AE = AD.\]

Step 2: Form a pair of equal ratios

Using these equalities:

\[\frac{AD}{AB} = \frac{AE}{AC}. \tag{1}\]

Step 3: Compare the included angle

The angle between \(AD\) and \(AE\) is \(\angle DAE\).

The angle between \(AB\) and \(AC\) is \(\angle BAC\).

Since D lies on \(AB\) and E lies on \(AC\), rays \(AD\) and \(AB\) are the same line, and rays \(AE\) and \(AC\) are the same line. Therefore,

\[\angle DAE = \angle BAC. \tag{2}\]

Step 4: Apply SAS similarity

From (1), the ratios of two pairs of corresponding sides are equal, and from (2), the included angle is equal. So, by the SAS similarity criterion:

\[\triangle ADE \sim \triangle ABC.\]

Hence proved.