State giving reasons, whether the following pairs of triangles are congruent or not: `DeltaKLM` in which KM=4 cm `angleK=75^(@),angleM=40^(@)` and `DeltaPQR` in which PR=4 cm, `angleQ=65^(@),angleR=40^(@)`.

Answer

To determine whether the triangles \( \Delta KLM \) and \( \Delta PQR \) are congruent, we will use the information provided and apply the criteria for triangle congruence.

### Step-by-Step Solution:

1. **Identify Given Information:**
- For triangle \( KLM \):
- \( KM = 4 \, \text{cm} \)
- \( \angle K = 75^\circ \)
- \( \angle M = 40^\circ \)
- For triangle \( PQR \):
- \( PR = 4 \, \text{cm} \)
- \( \angle Q = 65^\circ \)
- \( \angle R = 40^\circ \)

2. **Calculate the Missing Angle in Triangle \( PQR \):**
- The sum of angles in a triangle is \( 180^\circ \).
- For triangle \( PQR \):
\[ \angle P + \angle Q + \angle R = 180^\circ \]
\[ \angle P + 65^\circ + 40^\circ = 180^\circ \]
\[ \angle P + 105^\circ = 180^\circ \]
\[ \angle P = 180^\circ - 105^\circ = 75^\circ \]

3. **Compare Corresponding Angles:**
- Now we have:
- \( \angle K = 75^\circ \) (from \( KLM \))
- \( \angle P = 75^\circ \) (calculated for \( PQR \))
- \( \angle M = 40^\circ \) (from \( KLM \))
- \( \angle R = 40^\circ \) (from \( PQR \))

4. **Compare Corresponding Sides:**
- We also have:
- \( KM = 4 \, \text{cm} \) (from \( KLM \))
- \( PR = 4 \, \text{cm} \) (from \( PQR \))

5. **Establish Congruence:**
- We can now establish that:
- \( \angle K = \angle P \)
- \( \angle M = \angle R \)
- \( KM = PR \)
- By the Angle-Side-Angle (ASA) congruence criterion, since two angles and the included side of one triangle are equal to two angles and the included side of another triangle, we conclude that:
\[ \Delta KLM \cong \Delta PQR \]

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

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