# ABCD is a trapezium in which AB II CD. If angle ADC = twice angle ABC, AD = a cm and CD = b cm, then find the length of AB.

A trapezium is a quadrilateral in which one pair of opposite sides are parallel

## Answer: The length of AB is (a + b) cm.

Let's find the length of AB.

**Explanation:**

Let's draw the diagram of trapezium ABCD in which AB II CD

Here, ABCD is a trapezium with AB || CD.

⇒ angle ( ABC + ADC) = 180º

Given that, angle ADC = 2(angle ABC)

⇒ ABC + 2(ABC) = 180º

⇒ 3(ABC) = 180º

⇒ ABC = 180º/3 = 60º

Drop perpendiculars from the vertices D, C to AB, namely DE and CF. From the figure, DE is perpendicular to AB, then triangle ADE is formed with DAE = 60°.

cos60º = 1/2 = AE/AD

So, AE = AD/2 = a/2 and EF is equal to CD = b

AB = AE + EF + FB = a/2 + b + a/2 = (a/2 + a/2) + b = a + b

### Thus, the length of AB is (a + b) cm.

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