Question

segment addition: the sum of all parts equals the total.

\\( \overline { w x } \cong \overline { y z } \\), \\( x \\) is the midpoint of \\( \overline { w y } \\).

\\( x = \\) type your answer...

\\( x z = \\) type your answer...

Explanation:

Step1: Use the mid - point property

Since \(X\) is the mid - point of \(\overline{WY}\), \(WX = XY\). And given \(\overline{WX}\cong\overline{YZ}\), so \(WX = XY=YZ = 5x - 5\).
By segment addition, \(WZ=WX + XY+YZ\), and \(WZ = 6x + 30\).
So \(6x + 30=(5x - 5)+(5x - 5)+(5x - 5)\).

Step2: Simplify the equation

Simplify the right - hand side: \((5x - 5)+(5x - 5)+(5x - 5)=15x-15\).
The equation becomes \(6x + 30=15x-15\).
Subtract \(6x\) from both sides: \(30 = 15x-6x-15\), i.e., \(30=9x - 15\).
Add \(15\) to both sides: \(30 + 15=9x\), so \(45 = 9x\).
Divide both sides by \(9\): \(x=\frac{45}{9}=5\).

Step3: Find the length of \(XZ\)

\(XZ=XY + YZ\). Since \(XY = YZ=5x - 5\) and \(x = 5\), then \(XY=YZ=5\times5 - 5=20\).
So \(XZ=20 + 20=40\).

Answer:

\(x = 5\)
\(XZ=40\)