Parallels 'Parallel' to parallel

Method 1 :

$\text{In }\Delta\text{AGB and }\Delta\text{CGE} \\ \angle ABG = \angle CEG \text{ (Corresponding angles)} \\ \angle AGB = \angle CGE \text{ (Common angle)} \\ \implies \Delta AGB \sim \Delta CGE \text{ (By AA Similarity)} \\ \implies \dfrac{GE}{GB} = \dfrac{CE}{AB} \\ \implies \frac{GE}{GB} = \dfrac{z}{x} \text{ ... (i) }\\ \text{In }\Delta FBG, \Delta CBE \\ \angle FGB = \angle CEB \text{ (Corresponding angles)} \\ \angle FBG = \angle CBE \text{ (Common)} \\ \implies \Delta FBG \sim \Delta CBE \\ \implies \dfrac{BE}{BG} = \dfrac{CE}{FG} \\ \implies \dfrac{BE}{BG} = \frac{z}{y} \text{ ... (ii)}\\ (i) + (ii) \\ \implies \dfrac{z}{x} + \dfrac{z}{y} = \dfrac{GE}{GB} + \dfrac{BE}{GB} \\ \implies z (\dfrac{1}{x} + \dfrac{1}{y}) = \dfrac{GE + BE}{GB} = \dfrac{GB}{GB} \\ \implies z(\dfrac{1}{x} + \dfrac{1}{y}) = 1\\ \implies \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z} ... (iii)$

Method 2: $\text{From (iii)} \\ \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z} \\ \implies \dfrac{y+x}{xy} = \dfrac{1}{z} \\ \implies \dfrac{xy}{x+y} = z ... (iv)$

Substitute values of x, y as given in question in any of $(iii)$ or $(iv)$ , You'll get:

In $(iii)$ :

$\dfrac{1}{4.5} + \dfrac{1}{9} = \dfrac{1}{z} \\ \implies \dfrac{10}{45} + \dfrac{1}{9} = \dfrac{1}{z} \\ \implies \dfrac{10 + 5}{45} = \dfrac{1}{z} \\ \implies\dfrac{15}{45} = \dfrac{1}{z} \\ \implies \dfrac{1}{3} = \dfrac{1}{z} \\ \implies \fbox{z = 3}$

In $(iv)$ :

$\dfrac{xy}{x+y} = z\\ \implies \dfrac{4.5 \times 9}{4.5+ 9} = z \\ \implies \dfrac{40.5}{13.5} = z \\ \implies \dfrac{405}{135} = z \\ \implies \fbox{z = 3}$