If `triangleABC cong triangle FDE , AB = 5 cm , angleB = 40^(@) ,angle A = 80^(@)` Then which of the following is correct ?

Options

A. DF = 5 cm , `angleF = 60^(@`
B. DF = 5 cm ,`angleE = 60^(@)`
C. DF = 5 cm , `angle C = 60 ^(@)`
D. Both (B) and ( C)

Answer

To solve the problem step by step, we will analyze the information given about triangles ABC and FDE, which are congruent.

### Step 1: Understand the Given Information
We are given:
- Triangle ABC is congruent to triangle FDE.
- \( AB = 5 \, \text{cm} \)
- \( \angle B = 40^\circ \)
- \( \angle A = 80^\circ \)

### Step 2: Identify Corresponding Parts
Since the triangles are congruent, we can use the property of Corresponding Parts of Congruent Triangles (CPCT). This means:
- \( AB = FD \)
- \( \angle A = \angle F \)
- \( \angle B = \angle D \)
- \( \angle C = \angle E \)

### Step 3: Calculate the Remaining Angle in Triangle ABC
We can find \( \angle C \) using the angle sum property of triangles, which states that the sum of the angles in a triangle is \( 180^\circ \).

\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting the known values:
\[ 80^\circ + 40^\circ + \angle C = 180^\circ \]
\[ 120^\circ + \angle C = 180^\circ \]
\[ \angle C = 180^\circ - 120^\circ = 60^\circ \]

### Step 4: Find Corresponding Angles in Triangle FDE
Since the triangles are congruent, we have:
- \( \angle F = \angle A = 80^\circ \)
- \( \angle D = \angle B = 40^\circ \)
- \( \angle E = \angle C = 60^\circ \)

### Step 5: Verify the Lengths
From the congruence:
- Since \( AB = 5 \, \text{cm} \), it follows that \( FD = 5 \, \text{cm} \).

### Step 6: Analyze the Options
Now, we check the options provided:
1. \( DF = 5 \, \text{cm} \) and \( \angle F = 60^\circ \) (Incorrect, \( \angle F = 80^\circ \))
2. \( DF = 5 \, \text{cm} \) and \( \angle E = 60^\circ \) (Correct)
3. \( DF = 5 \, \text{cm} \) and \( \angle C = 60^\circ \) (Correct)
4. Both B and C (Correct)

### Final Answer
Thus, the correct answer is option D: Both B and C.

---