Question

In the given figure, \(\Delta ABE \cong \Delta ACD\). Prove that \(\Delta ADE \sim \Delta ABC\).

Hint:

Whenever you need to prove similarity inside a larger triangle where parts are congruent, look for a shared angle.
The shared angle \( \angle A \) is almost always the key.
Establishing the proportional relation of the adjacent sides \( AD/AB = AE/AC \) immediately allows the use of the SAS similarity criterion.

Answer 1

Step 1: Locate D and E on the two sides of triangle ABC.
From the figure, D lies on side AB and E lies on side AC. Because of this, ray AD is the same ray as ray AB, and ray AE is the same ray as ray AC.
This means the angle between AD and AE is not merely equal to the angle between AB and AC, it is literally the very same angle, angle A.
\[ \angle DAE = \angle BAC \quad \text{(the same angle, shared by both triangles)} \]

Step 2: Use the given congruence to get equal sides.
We are given $\Delta ABE \cong \Delta ACD$, with the correspondence $A \leftrightarrow A$, $B \leftrightarrow C$, $E \leftrightarrow D$.
By CPCT (corresponding parts of congruent triangles are equal):
\[ AB = AC \quad \text{(i)} \]
\[ AE = AD \quad \text{(ii)} \]

Step 3: Notice what this says about points D and E.
Since $AB = AC$ and $AD = AE$, subtracting equation (ii) from equation (i) gives:
\[ AB - AD = AC - AE \]
\[ DB = EC \]
This confirms D and E sit at matching positions on AB and AC, consistent with the figure. We do not need this fact directly for the similarity proof, but it is a useful check on the setup.

Step 4: Form the ratio of sides needed for SAS similarity.
Divide equation (ii) by equation (i), which is valid since $AB = AC \neq 0$:
\[ \frac{AD}{AB} = \frac{AE}{AC} \]

Step 5: Apply the SAS similarity criterion.
In triangles ADE and ABC, we now have two sides in the same ratio, $\frac{AD}{AB} = \frac{AE}{AC}$, and the angle included between those two sides is exactly the same angle in both triangles, angle A, as shown in Step 1.
This is exactly what the Side Angle Side similarity criterion requires, so:
\[ \Delta ADE \sim \Delta ABC \]

Final Answer:
Hence, it is proved that $\Delta ADE \sim \Delta ABC$ by the SAS similarity criterion.