Find the unknown values in each of the following figures. All lengths are given in centimeters. (All measures are not in scale)

In this question we must find the value of x and y, i.e. AC and AG respectively.

In such questions where we are given with some sides or angles of one or more triangles and we need find the other remaining sides and angles of triangles. The concept generally used is of congruency and similarity.

In this question we will be dealing only with sides.

Here,

In ΔADE and ΔABC

∠A is common

∠ACB = ∠AED by correspondence(BC∥DE),

∠ABC = ∠ADE by correspondence(FC∥DA)

∴ ΔADE ∼ ΔABC

Similarly, ΔEGA∼ΔEFC

(∵ ΔADE∼ΔABC)

⇒ 24x = 8x + 64

⇒ 16x = 64

⇒ x = 4 cm

Also,

(∵ΔEGA∼ΔEFC)

(∵EA = 8 + x, GA = y)

⇒ y = 9 cm

⇒ y = 4 cm