In the figure below, arc UVW = arc WXY and UVW = 110 degrees. Find: a) angle WUX, b) angle WUX, c) angle XWY, d) angle YXW

• Pembe $\angle UVW = 110^\circ$ ni pembe iliyoandikwa kwenye duara. Inaunda arc UW (arc kubwa isiyo na V). Hivyo, $m(UW)_{\text{kubwa}} = 2 \times \angle UVW = 2 \times 110^\circ = 220^\circ$.
• Arc ndogo UW (inayopitia V) ni $360^\circ - 220^\circ = 140^\circ$. Hivyo, $m(UVW) = 140^\circ$.
• Tunapewa $arc(UVW) = arc(WXY)$. Hivyo, $m(WXY) = 140^\circ$.
• Tunapewa Chord WX = Chord XY. Hii inamaanisha arc zao zinazofanana ni sawa: $m(WX) = m(XY)$.
• Kutoka $m(WXY) = m(WX) + m(XY)$, tunapata: $140^\circ = m(WX) + m(WX) = 2 \times m(WX)$. Hivyo, $m(WX) = \frac{140^\circ}{2} = 70^\circ$. Na $m(XY) = 70^\circ$.
• Jumla ya arc zote kwenye duara ni $360^\circ$. $m(UV) + m(VW) + m(WX) + m(XY) + m(YU) = 360^\circ$. Tunajua $m(UV) + m(VW) = m(UVW) = 140^\circ$. Hivyo, $140^\circ + 70^\circ + 70^\circ + m(YU) = 360^\circ$. $280^\circ + m(YU) = 360^\circ$. $m(YU) = 360^\circ - 280^\circ = 80^\circ$.

Sasa tuna vipimo vya arc:
$m(WX) = 70^\circ$
$m(XY) = 70^\circ$
$m(YU) = 80^\circ$
$m(UVW) = 140^\circ$

a) $\angle WUY$
Pembe $\angle WUY$ inaunda arc WY.
$m(WY) = m(WX) + m(XY) = 70^\circ + 70^\circ = 140^\circ$.

$$\angle WUY = \frac{1}{2} m(WY) = \frac{1}{2} \times 140^\circ = 70^\circ$$ $$\boxed{\angle WUY = 70^\circ}$$

b) $\angle WUX$
Pembe $\angle WUX$ inaunda arc WX.

$$\angle WUX = \frac{1}{2} m(WX) = \frac{1}{2} \times 70^\circ = 35^\circ$$ $$\boxed{\angle WUX = 35^\circ}$$

c) $\angle XWY$
Pembe $\angle XWY$ inaunda arc XY.

$$\angle XWY = \frac{1}{2} m(XY) = \frac{1}{2} \times 70^\circ = 35^\circ$$ $$\boxed{\angle XWY = 35^\circ}$$

**d) $\angle YXW** Pembe$\angle YXW$inaunda **arc** YW (arc isiyo na X).$m(YW) = m(YU) + m(UVW) = 80^\circ + 140^\circ = 220^\circ$.

$$\angle YXW = \frac{1}{2} m(YW) = \frac{1}{2} \times 220^\circ = 110^\circ$$ $$\boxed{\angle YXW = 110^\circ}$$