Question

given △stu with st || xy. prove \\( \frac { s x } { x u } = \frac { t y } { y u } \\). complete the steps of the proof. triangle stu is similar to triangle xyu. substitution property. subtraction property. transitive property. statements: 1 △stu with st || xy. 2 ∠1 and ∠2 are corresponding angles. 3 ∠3 and ∠4 are corresponding angles. 4 ∠1 ≅ ∠2, ∠3 ≅ ∠4. 5. 6 \\( \frac { s u } { x u } = \frac { t u } { y u } \\). 7 su = sx + xu; tu = ty + yu. 8 \\( \frac { s x + x u } { x u } = \frac { t y + y u } { y u } \\). reasons: 1 given. 2 def of corresponding angles. 3 def of corresponding angles. 4 corresponding angles theorem. 5 aa similarity theorem. 6 def of similar triangles. 7 segment addition postulate. 8 substitution property.

Explanation:

Step1: Apply AA similarity theorem

Since \(\angle1\cong\angle2\) and \(\angle3\cong\angle4\) (from steps 4), by AA (Angle - Angle) similarity theorem, \(\triangle XYU\sim\triangle STU\).

Step2: Use the property of similar triangles

For similar triangles \(\triangle XYU\) and \(\triangle STU\), we have \(\frac{SU}{XU}=\frac{TU}{YU}\) (step 6).

Step3: Substitute segment addition

Given \(SU = SX+XU\) and \(TU=TY + YU\) (step 7), substitute into \(\frac{SU}{XU}=\frac{TU}{YU}\) to get \(\frac{SX + XU}{XU}=\frac{TY+YU}{YU}\).

Step4: Use subtraction property

By the subtraction property of equality, if \(\frac{a + b}{b}=\frac{c + d}{d}\), then \(\frac{a}{b}=\frac{c}{d}\). Here \(a = SX\), \(b=XU\), \(c = TY\), \(d = YU\). So \(\frac{SX}{XU}=\frac{TY}{YU}\) (from step 8, after applying subtraction property to \(\frac{SX + XU}{XU}=\frac{TY+YU}{YU}\)).

Answer:

\(\frac{SX}{XU}=\frac{TY}{YU}\)