Answer:
A) UT = 42; WT = 21; ST = 42
[tex]\text{d) } m\widehat {XT} = 56^{\circ}; \text{e) } m\widehat {ST} = 112^{\circ}; \text{f) }m\widehat {US} = 136^{\circ}[/tex]
Step-by-step explanation:
a), b), and c)
YZ ⟂ ST, so SV = TV = 21
In ∆s ZVT and ZWT,
∠ZVT = ∠ZWT; ZV = ZW; ZT is common.
∴ ∆ZVT ≅ ∆ZWT
∴ ∠ZTV = ∠ZTW
In ∆s SVZ and TVZ,
SV = TV; SZ = TZ; VZ is common
∴ ∆SVZ ≅ ∆TVZ
∴ ∠SZV = ∠TZV
By similar reasoning,
∆TWZ ≅ ∆UWZ
∴ ∠TZW = ∠UZW
So, the four angles marked with red dots are equal.
Also, SV = TV = TW = UW = 21
In ∆s STZ and UTZ,
SZ = UZ; ST =UT; TZ is common
∴ ∆STZ ≅ ∆UTZ
∴ ∠SZT = ∠UZT and
ST = UT = 42
[tex]\textbf{e) m} \mathbf{\widehat {ST}}\\m \widehat {ST} =m \widehat {UT } = 112^{\circ}[/tex]
[tex]\textbf{d) m} \mathbf{\widehat {XT}}\\m \widehat {XT} = \frac{1}{2} m \widehat {UT } = \frac{1}{2}(112^{\circ}) = \mathbf{56^{\circ}}[/tex]
[tex]\textbf{f) m} \mathbf{\widehat {US}}\\m\widehat {US} =360^{\circ} - 2\times 112^{\circ} = 360^{\circ} - 224^{\circ} = \mathbf{136^{\circ}}[/tex]
