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: Identifying the Triangles:
We compare triangles AEF and ABC.
To prove similarity, we use the AA similarity criterion.

Step 2: Common Angle:
∠EAF = ∠BAC
This is the same angle at vertex A for both triangles.

Step 3: Using Parallel Lines:
Given EF ∥ BC.

When a transversal cuts parallel lines, alternate interior angles are equal.

So,
∠AEF = ∠ABC
∠AFE = ∠ACB

Step 4: Applying AA Similarity:
In triangles AEF and ABC:
One angle is common (∠A).
Another pair of corresponding angles are equal due to parallel lines.

Therefore,
ΔAEF ∼ ΔABC
(by AA similarity criterion)

Final Answer:
Triangles AEF and ABC are similar by AA criterion since ∠A is common and EF ∥ BC gives equal corresponding angles.