Question

In the same figure as 32(a), prove that \(\triangle AEF \sim \triangle ABC\).

Hint:

Parallel lines within a triangle always create a smaller triangle similar to the original one.

Answer 1

Step 1: Understanding the Concept:
Two triangles are similar if their corresponding angles are equal (AA similarity criterion).
Step 2: Detailed Explanation:
1. In \(\triangle AEF\) and \(\triangle ABC\):
- \(\angle EAF = \angle BAC\) (Common angle for both triangles).
2. From part (ii), we proved that \(EF \parallel BC\).
3. When parallel lines are intersected by a transversal, corresponding angles are equal.
- \(\angle AEF = \angle ABC\) (Corresponding angles).
- \(\angle AFE = \angle ACB\) (Corresponding angles).
4. Since two corresponding angles are equal, the triangles are similar by the AA (Angle-Angle) similarity criterion.
\[ \triangle AEF \sim \triangle ABC \]
Step 3: Final Answer:
The similarity is proven by the AA criterion because \(\angle A\) is common and \(\angle AEF = \angle ABC\) due to \(EF \parallel BC\).