Question

if \\( \angle rsu \cong \angle tsu \\), \\( ru = 10w + 19 \\), and \\( tu = w + 91 \\), what is \\( tu \\)?

Explanation:

Step1: Use the Angle - Bisector Theorem

Since \( \angle RSU\cong\angle TSU\), and \(SR\perp RU\), \(ST\perp TU\), by the Angle - Bisector Theorem (a point on the bisector of an angle is equidistant from the sides of the angle), we have \(RU = TU\).

Step2: Set up the equation

Set \(10w + 19=w + 91\).
Subtract \(w\) from both sides: \(10w - w+19=w - w + 91\), which simplifies to \(9w+19 = 91\).
Subtract 19 from both sides: \(9w+19 - 19=91 - 19\), so \(9w=72\).
Divide both sides by 9: \(w=\frac{72}{9}=8\).

Step3: Find the value of \(TU\)

Substitute \(w = 8\) into the formula for \(TU\). Since \(TU=w + 91\), then \(TU=8 + 91\).

Answer:

\(99\)