Question

given: \\( \triangle stu \\) with \\( \overline{st} \parallel \overline{xy} \\)
prove: \\( \frac{sx}{xu}=\frac{ty}{yu} \\)
complete the steps of the proof.

Explanation:

Step1: Identify similar triangles

Since \( \angle1\cong\angle2\) and \( \angle3\cong\angle4\), by AA (Angle - Angle) similarity theorem, \( \triangle UXY\sim\triangle UST\).

Step2: Analyze step 11

In step 10, we have \( \frac{SX}{XU}+1=\frac{TY}{YU}+1\). Subtracting 1 from both sides of the equation \(a + 1=b + 1\) (where \(a=\frac{SX}{XU}\) and \(b = \frac{TY}{YU}\)) gives the result. According to the subtraction property of equality: if \(a + c=b + c\), then \(a=b\). Here \(c = 1\), \(a=\frac{SX}{XU}\), and \(b=\frac{TY}{YU}\).

Answer:

♠: \( \triangle UXY\sim\triangle UST\)
♦: subtraction property of equality