Question

solve for x. round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.
two triangles: left triangle tsr with ts=19, sr=25, angles at t: 58°, at s: 81°; right triangle vwu with vw=46, angle at v: 81°, angle at u: 41°, side vu=x

Explanation:

Step1: Check triangle similarity

First, find the third angle of each triangle. For triangle \(TSR\): \(\angle R = 180^\circ - 58^\circ - 81^\circ = 41^\circ\). For triangle \(VWU\): \(\angle W = 180^\circ - 81^\circ - 41^\circ = 58^\circ\). So, \(\triangle TSR \sim \triangle VWU\) by AA similarity (two angles equal).

Step2: Set up proportion

Corresponding sides: \(TS = 19\), \(VW = 46\), \(SR = 25\), \(VU = x\). So, \(\frac{TS}{VW}=\frac{SR}{VU}\) (since corresponding sides of similar triangles are proportional).

Step3: Solve for \(x\)

Substitute values: \(\frac{19}{46}=\frac{25}{x}\). Cross - multiply: \(19x = 46\times25\). Calculate \(46\times25 = 1150\). Then \(x=\frac{1150}{19}\approx60.5\).

Answer:

\(x\approx60.5\)