Segment AB is parallel to segment CD. AD intersects BC in E. If lengths of AE, BC and ED are 10 cm, 15 cm and 15 cm, what is the length of EC?

Given : AB is parallel to CD. AE = 10 cm ,  BC = 15 cm and ED = 15 cm

To find : EC = ?

Solution : Let $$EC = x$$ cm

In $$\triangle$$ BAE and $$\triangle$$ CDE

=> $$\angle$$ ABE = $$\angle$$ DCE    (Alternate interior angles)

=> $$\angle$$ BAE = $$\angle$$ CDE    (Alternate interior angles)

=> $$\angle$$ AEB = $$\angle$$ CED    (Vertically opposite angles)

Thus, $$\triangle$$ BAE $$\sim$$ $$\triangle$$ CDE  (AAA criteria)

$$\therefore$$ $$\frac{AE}{DE} = \frac{EB}{EC}$$

=> $$\frac{10}{15}=\frac{(15-x)}{x}$$

=> $$\frac{(15-x)}{x}=\frac{2}{3}$$

=> $$45-3x=2x$$

=> $$2x+3x=5x=45$$

=> $$x=\frac{45}{5}=9$$ cm

=> Ans - (D)