Show that triangle ABC in Fig. 11.23 is similar to triangle LMN in Fig. 11.22 below. Hence find the values of x and y.

Jumla ya pembe katika pembetatu ni $180^\circ$.

• Kwa pembetatu LMN (Mchoro 11.22): Pembe L = $55^\circ$ Pembe M = $75^\circ$ Pembe N = $180^\circ - (55^\circ + 75^\circ) = 180^\circ - 130^\circ = 50^\circ$

• Kwa pembetatu ABC (Mchoro 11.23): Pembe B = $75^\circ$ Pembe C = $50^\circ$ Pembe A = $180^\circ - (75^\circ + 50^\circ) = 180^\circ - 125^\circ = 55^\circ$

• Pembe L = $55^\circ$ na Pembe A = $55^\circ$
• Pembe M = $75^\circ$ na Pembe B = $75^\circ$
• Pembe N = $50^\circ$ na Pembe C = $50^\circ$

Kwa kuwa pembe zote zinazolingana ni sawa, pembetatu $\triangle LMN$ na $\triangle ABC$ zinafanana (kwa kigezo cha AAA - Angle-Angle-Angle).
Hivyo, $\triangle LMN \sim \triangle ABC$.

Sehemu ya 2: Kutafuta thamani za $x$ na $y$

Kwa kuwa pembetatu hizo zinafanana, uwiano wa pande zao zinazolingana ni sawa.
Pande zinazolingana ni: LM na AB, MN na BC, LN na AC.

$$\frac{LM}{AB} = \frac{MN}{BC} = \frac{LN}{AC}$$

Tumia uwiano wa kwanza na wa pili:

$$\frac{2}{5} = \frac{y}{7.5}$$

Kuzidisha msalaba:

$$5y = 2 \times 7.5$$ $$5y = 15$$ $$y = \frac{15}{5}$$ $$y = 3$$

Tumia uwiano wa kwanza na wa tatu:

$$\frac{2}{5} = \frac{4}{x}$$

Kuzidisha msalaba:

$$2x = 5 \times 4$$ $$2x = 20$$ $$x = \frac{20}{2}$$ $$x = 10$$

Jibu:
Thamani za $x$ na $y$ ni:
$x = \boxed{10 \text{ cm}}$
$y = \boxed{3 \text{ cm}}$