Question

In ΔABC, ∠B = 50°, ∠ADB = 80°, AD = DC.

∠C =

पर्याय

• 50°

• 40°

• 45°

• 60°

Answer 1

40°

Explanation:

Consider quadrilateral ADBC. Given: AD = DC and ∠ADB = 80°. Since AD = DC, triangle ADC is isosceles with ∠ADC = ∠DAC.

Let ∠DAC = ∠ADC = x.

Because quadrilateral ADBC is cyclic (the points lie on a circle), ∠B + ∠ADC = 180° (opposite angles of a cyclic quadrilateral sum to 180°). Given ∠B = 50°.

So,

∠ADC = 180° − ∠B = 180° − 50° = 130°

But this contradicts ∠ADC = x and triangle ADC being isosceles with angles x, x, and 80°, because 80° + 2x = 180°, which gives x = 50°.

Therefore, the assumption that quadrilateral is cyclic is incorrect.

Instead, by considering triangle ABD and triangle CDB with AD = DC and using the exterior angle theorem and angle sum properties, the angle ∠C = 40°.

Therefore, the answer is 40°.

Hence, ∠C = 40°.