For which of the following measurements, the construction of triangle ABC can be done ?

The correct option is AB = 8 cm, BC = 4 cm, ∠ BCA=90°, ∠ ABC =60°.

To determine for which set of measurements a triangle ABC can be constructed, let us analyze the geometric properties required for a valid triangle for each given option:

1. Checking Option 1:
Measurements given: AB = 5 cm, BC = 6 cm, ∠ ABC = 130°, ∠ BAC = 25°.

The sum of the two given angles is:

$\angle ABC+\angle BAC=130°+25°=155°$

The third angle would be:

$\angle ACB=180°-155°=25°$

In a triangle, sides opposite to equal angles must be equal. Since ∠ BAC = ∠ ACB = 25°, the triangle must be isosceles with BC = AB. However, the given values state AB = 5 cm and BC = 6 cm, which contradicts BC = AB. Thus, this triangle cannot be constructed with these contradictory dimensions.

2. Checking Option 2:
Measurements given: AB = 3 cm, BC = 4 cm, AC = 7 cm.

According to the triangle inequality theorem, the sum of any two sides of a triangle must be strictly greater than the third side. Here, the sum of two smaller sides is:

$AB+BC=3\text{cm}+4\text{cm}=7\text{cm}$

Since AB + BC = AC (7 cm = 7 cm), the points A, B, and C lie on a single straight line (collinear) and do not form a triangle. Thus, construction is impossible.

3. Checking Option 3:
Measurements given: AB = 8 cm, BC = 4 cm, ∠ BCA = 90°, ∠ ABC = 60°.

In a right-angled triangle ABC right-angled at C (∠ BCA = 90°), the hypotenuse is the side opposite to the right angle, which is AB. By trigonometric ratio rules for a right triangle with acute angle ∠ ABC = 60°:

$\cos (\angle ABC)=\frac{\text{Adjacent Side}}{\text{Hypotenuse}}=\frac{BC}{AB}$

Substituting the values:

$\cos (60°)=\frac{4}{8}=\frac{1}{2}$

Since $\cos (60°)=\frac{1}{2}$ is mathematically correct, all given angle and side relationships are consistent. Therefore, triangle ABC can be successfully constructed.

4. Checking Option 4:
Measurements given: ∠ ACB = 120°, ∠ ABC = 25°, AC = BC = 4 cm.

Since AC = BC, triangle ABC is isosceles with base AB, meaning the base angles opposite to these equal sides must be equal: ∠ CAB = ∠ ABC = 25°.

The sum of all interior angles of the triangle would be:

$\angle ACB+\angle ABC+\angle CAB=120°+25°+25°=170°\ne 180°$

Since the sum of interior angles does not equal 180°, this triangle cannot exist.

Hence, the construction of triangle ABC is possible only for AB = 8 cm, BC = 4 cm, ∠ BCA=90°, ∠ ABC =60°.