In circle Z, if vz = zw, sv=21, and mUT=112degrees, find each measure

A) UT = 42; WT = 21; ST = 42

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