The diagram below is a quadrilateral ABCD in which CD = 8 cm, BC = 11cm, AC = 13cm, AB = 4cm and angle ADC = 80°.

$$\frac{\sin(\angle CAD)}{CD} = \frac{\sin(\angle ADC)}{AC}$$ $$\frac{\sin(\angle CAD)}{8} = \frac{\sin 80^\circ}{13}$$ $$\sin(\angle CAD) = \frac{8 \sin 80^\circ}{13}$$ $$\sin(\angle CAD) = \frac{8 \times 0.9848}{13}$$ $$\sin(\angle CAD) \approx 0.6060$$ $$\angle CAD = \arcsin(0.6060) \approx 37.3^\circ$$

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

$$\angle ACD = 180^\circ - \angle ADC - \angle CAD$$ $$\angle ACD = 180^\circ - 80^\circ - 37.3^\circ$$ $$\angle ACD = 62.7^\circ$$

Ukubwa wa pembe ACD ni $\boxed{\text{62.7}^\circ}$.

$$AC^2 = AD^2 + CD^2 - 2(AD)(CD)\cos(\angle ADC)$$ $$13^2 = AD^2 + 8^2 - 2(AD)(8)\cos(80^\circ)$$ $$169 = AD^2 + 64 - 16 AD (0.1736)$$ $$169 = AD^2 + 64 - 2.7776 AD$$

$$AD^2 - 2.7776 AD + 64 - 169 = 0$$ $$AD^2 - 2.7776 AD - 105 = 0$$

Tumia formula ya quadratic $AD = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:

$$AD = \frac{2.7776 \pm \sqrt{(-2.7776)^2 - 4(1)(-105)}}{2(1)}$$ $$AD = \frac{2.7776 \pm \sqrt{7.715 + 420}}{2}$$ $$AD = \frac{2.7776 \pm \sqrt{427.715}}{2}$$ $$AD = \frac{2.7776 \pm 20.681}{2}$$

Kwa kuwa urefu hauwezi kuwa hasi, tunachukua thamani chanya:

$$AD = \frac{2.7776 + 20.681}{2}$$ $$AD = \frac{23.4586}{2}$$ $$AD \approx 11.7293 \text{ cm}$$

Urefu wa upande AD ni $\boxed{\text{11.7 cm}}$.

$$AB^2 = BC^2 + AC^2 - 2(BC)(AC)\cos(\angle ACB)$$ $$4^2 = 11^2 + 13^2 - 2(11)(13)\cos(\angle ACB)$$ $$16 = 121 + 169 - 286 \cos(\angle ACB)$$ $$16 = 290 - 286 \cos(\angle ACB)$$

$$286 \cos(\angle ACB) = 290 - 16$$ $$286 \cos(\angle ACB) = 274$$ $$\cos(\angle ACB) = \frac{274}{286}$$ $$\cos(\angle ACB) \approx 0.9580$$ $$\angle ACB = \arccos(0.9580) \approx 16.7^\circ$$

Ukubwa wa pembe ACB ni $\boxed{\text{16.7}^\circ}$.

Tunajua $CD = 8$ cm, $AD \approx 11.7293$ cm, na $\angle ADC = 80^\circ$.

$$\text{Eneo}(\triangle ADC) = \frac{1}{2} \times CD \times AD \times \sin(\angle ADC)$$ $$\text{Eneo}(\triangle ADC) = \frac{1}{2} \times 8 \times 11.7293 \times \sin(80^\circ)$$ $$\text{Eneo}(\triangle ADC) = 4 \times 11.7293 \times 0.9848$$ $$\text{Eneo}(\triangle ADC) \approx 46.20 \text{ cm}^2$$

Tunajua $BC = 11$ cm, $AC = 13$ cm, na $\angle ACB \approx 16.664^\circ$.

$$\text{Eneo}(\triangle ABC) = \frac{1}{2} \times BC \times AC \times \sin(\angle ACB)$$ $$\text{Eneo}(\triangle ABC) = \frac{1}{2} \times 11 \times 13 \times \sin(16.664^\circ)$$ $$\text{Eneo}(\triangle ABC) = \frac{1}{2} \times 143 \times 0.2866$$ $$\text{Eneo}(\triangle ABC) \approx 20.49 \text{ cm}^2$$

$$\text{Eneo(ABCD)} = \text{Eneo}(\triangle ADC) + \text{Eneo}(\triangle ABC)$$ $$\text{Eneo(ABCD)} = 46.20 + 20.49$$ $$\text{Eneo(ABCD)} = 66.69 \text{ cm}^2$$

Eneo la quadrilateral ABCD ni $\boxed{\text{66.7 cm}^2}$.