In triangles ABD and FEC, ∠BAD = 60°, BD = EC, ∠ABD = ∠FEC = 90°, and AB = FE. Find the ratio of ∠BAD to ∠FCE.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1746991335/aptidude_questions/mejto23ej3hf8kocicz3.png)

Question

In triangles ABD and FEC, ∠BAD = 60°, BD = EC, ∠ABD = ∠FEC = 90°, and AB = FE. Find the ratio of ∠BAD to ∠FCE.

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1746991335/aptidude_questions/mejto23ej3hf8kocicz3.png)

Accepted Answer

Given the triangles $$ 	riangle ABD $$ and $$ 	riangle FEC $$, we have:

- $$ \angle BAD = 60^\circ $$
- $$ BD = EC $$
- $$ \angle ABD = \angle FEC = 90^\circ $$
- $$ AB = FE $$

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### Step 1: Understand the triangles

- Both triangles are right-angled (at $$ B $$ and $$ E $$ respectively).
- $$ BD = EC $$ means the legs opposite the right angles in both triangles are equal.
- $$ AB = FE $$ means the other legs adjacent to the right angle are equal.
- $$ \angle BAD = 60^\circ $$ is given.

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### Step 2: Use the properties of right triangles

In right triangles $$ 	riangle ABD $$ and $$ 	riangle FEC $$:

- Since $$ BD = EC $$ and $$ AB = FE $$, the triangles are congruent by RHS (Right angle-Hypotenuse-Side) criterion.
- Therefore, corresponding angles are equal except the angle $$ \angle BAD $$ and $$ \angle FCE $$ which we want to find the ratio of.

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### Step 3: Find $$ \angle FCE $$

- Since $$ \angle ABD = 90^\circ $$, $$ \angle BAD = 60^\circ $$, the remaining angle $$ \angle ADB = 30^\circ $$ (sum of angles in a triangle is 180°).
- Similarly, in $$ 	riangle FEC $$, $$ \angle FEC = 90^\circ $$, and the other two angles must add to 90°.
- The angle $$ \angle FCE $$ corresponds to $$ \angle ADB $$ but is not equal; we need to find the ratio.

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### Step 4: Calculate the ratio $$ \frac{\angle BAD}{\angle FCE} $$

- Given $$ \angle BAD = 60^\circ $$.
- From the figure and congruence, $$ \angle FCE = 30^\circ $$.

Therefore,  
$$
\frac{\angle BAD}{\angle FCE} = \frac{60^\circ}{30^\circ} = 2 : 1
$$

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### Final Answer:
**4. 2 : 1**

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