In a Δ ABC, (c + a + b)(a + b - c) = ab. The measure of the angle C is:

Concept:

Cosine Rule:

For a Δ ABC, with sides a, b, c opposite to the angles A, B, C:

• a² = b² + c² - 2bc cos A.
• b² = a² + c² - 2ac cos B.
• c² = a² + b² - 2ab cos C.

Trigonometric Ratios for Allied Angles: 

• sin (-θ) = -sin θ.
• cos (-θ) = cos θ.
• sin (nπ + θ) = (-1)n sin θ.
• cos (nπ + θ) = (-1)n cos θ.
• \(\rm \sin \left [(2n+1)\dfrac{\pi}{2}+\theta \right ]\) = (-1)n cos θ.
• \(\rm \cos \left [(2n+1)\dfrac{\pi}{2}+\theta \right ]\) = (-1)n (-sin θ).

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Calculation:

It is given that (a + b + c)(a + b - c) = ab

⇒ (a + b)² - c² = ab

⇒ c² = (a + b)² - ab

⇒ c² = a² + b² + ab

Using the cosine rule:

⇒ a2 + b2 - 2ab cos C = a2 + b2 + ab

⇒ \(\rm \cos C=-\dfrac{1}{2}\)

⇒ \(\rm \cos C=-\sin \dfrac{\pi}{6}\)

⇒ \(\rm C= (2n+1)\dfrac{\pi}{2}+\dfrac{\pi}{6}\), where n is even.

∵ 0 < C < π, n must be 0.

⇒ \(\rm C= \dfrac{\pi}{2}+\dfrac{\pi}{6}=\dfrac{2\pi}{3}\).