In Fig. 6.5, if \(\angle D = \angle C\), is it true that \(\triangle ADE \sim \triangle ACB\)? Why?
Yes.
Step 1: Look at both triangles: \(\triangle ADE\) and \(\triangle ACB\).
Step 2: Notice that \(\angle A\) is present in both triangles. So, they share a common angle.
Step 3: It is given that \(\angle D = \angle C\).
Step 4: Now each triangle has two angles equal:
• In \(\triangle ADE\): angles \(\angle A\) and \(\angle D\).
• In \(\triangle ACB\): angles \(\angle A\) and \(\angle C\).
Step 5: When two angles of one triangle are equal to two angles of another triangle, the two triangles are similar. This rule is called the AA (Angle–Angle) similarity criterion.
Conclusion: Therefore, \(\triangle ADE \sim \triangle ACB\).