State giving reasons, whether the following pairs of triangles are congruent or not: `DeltaABC` in which `angleA=50^(@),angleB=60^(@),BC=4.5` cm and `DeltaDEF`o in which `angleE=60^(@),EF=4.5cm,angleF=70^(@)`

Answer

To determine whether the triangles \( \Delta ABC \) and \( \Delta DEF \) are congruent, we will analyze the given information step by step.

### Given:
- Triangle \( \Delta ABC \):
- \( \angle A = 50^\circ \)
- \( \angle B = 60^\circ \)
- \( BC = 4.5 \) cm

- Triangle \( \Delta DEF \):
- \( \angle E = 60^\circ \)
- \( \angle F = 70^\circ \)
- \( EF = 4.5 \) cm

### Step 1: Calculate the third angle of each triangle.
For triangle \( \Delta ABC \):
- The sum of angles in a triangle is \( 180^\circ \).
- Therefore, \( \angle C = 180^\circ - \angle A - \angle B \)
- \( \angle C = 180^\circ - 50^\circ - 60^\circ = 70^\circ \)

For triangle \( \Delta DEF \):
- Similarly, \( \angle D = 180^\circ - \angle E - \angle F \)
- \( \angle D = 180^\circ - 60^\circ - 70^\circ = 50^\circ \)

### Step 2: List the angles and sides of both triangles.
- For \( \Delta ABC \):
- \( \angle A = 50^\circ \)
- \( \angle B = 60^\circ \)
- \( \angle C = 70^\circ \)
- \( BC = 4.5 \) cm

- For \( \Delta DEF \):
- \( \angle D = 50^\circ \)
- \( \angle E = 60^\circ \)
- \( \angle F = 70^\circ \)
- \( EF = 4.5 \) cm

### Step 3: Compare the angles and sides.
- We have:
- \( \angle A = \angle D = 50^\circ \)
- \( \angle B = \angle E = 60^\circ \)
- \( \angle C = \angle F = 70^\circ \)
- \( BC = EF = 4.5 \) cm

### Step 4: Apply the Angle-Angle-Side (AAS) Congruence Criterion.
Since we have:
- Two angles and the included side of one triangle equal to the corresponding two angles and the included side of the other triangle, we can conclude that the two triangles are congruent.

### Conclusion:
Thus, \( \Delta ABC \cong \Delta DEF \) by the AAS criterion.

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