State giving reasons, whether the following pairs of triangles are congruent or not: `DeltaABC` in which AB=3 cm `angleA=90^(@)`, BC=5 cm and `DeltaKLM` in which KM=3 cm `angleK=90^(@)` cm, 1,M=5cm

Answer

To determine whether the triangles \( \Delta ABC \) and \( \Delta KLM \) are congruent, we will follow these steps:

### Step 1: Identify the given information
- For triangle \( \Delta ABC \):
- \( AB = 3 \, \text{cm} \)
- \( \angle A = 90^\circ \)
- \( BC = 5 \, \text{cm} \)

- For triangle \( \Delta KLM \):
- \( KM = 3 \, \text{cm} \)
- \( \angle K = 90^\circ \)
- \( LM = 5 \, \text{cm} \)

### Step 2: Use the Pythagorean theorem to find the missing side
Since both triangles are right-angled triangles, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

For triangle \( \Delta ABC \):
- Let \( AC \) be the missing side.
- According to the Pythagorean theorem:
\[ AC^2 + AB^2 = BC^2 \]
Substituting the known values:
\[ AC^2 + 3^2 = 5^2 \]
\[ AC^2 + 9 = 25 \]
\[ AC^2 = 25 - 9 = 16 \]
\[ AC = \sqrt{16} = 4 \, \text{cm} \]

For triangle \( \Delta KLM \):
- Let \( KL \) be the missing side.
- Using the Pythagorean theorem:
\[ KL^2 + KM^2 = LM^2 \]
Substituting the known values:
\[ KL^2 + 3^2 = 5^2 \]
\[ KL^2 + 9 = 25 \]
\[ KL^2 = 25 - 9 = 16 \]
\[ KL = \sqrt{16} = 4 \, \text{cm} \]

### Step 3: Compare the sides of the triangles
Now we have the lengths of all sides for both triangles:
- For \( \Delta ABC \): \( AB = 3 \, \text{cm}, AC = 4 \, \text{cm}, BC = 5 \, \text{cm} \)
- For \( \Delta KLM \): \( KM = 3 \, \text{cm}, KL = 4 \, \text{cm}, LM = 5 \, \text{cm} \)

### Step 4: State the congruence
Since all corresponding sides of the triangles are equal:
- \( AB = KM \)
- \( AC = KL \)
- \( BC = LM \)

By the Side-Side-Side (SSS) congruence criterion, we can conclude that:
\[ \Delta ABC \cong \Delta KLM \]

### Conclusion
The triangles \( \Delta ABC \) and \( \Delta KLM \) are congruent.

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