## Geometry: Common Core (15th Edition)

$TW = 10$
The converse of the angle bisector theorem states that a point inside an angle that is equidistant from the two sides of the angle is located on the bisector of that angle. In this diagram, we see that $\overline{WY}$ is equidistant from the two sides of $\angle TWZ$, so $\overline{WY}$ is the angle bisector of $\angle TWZ$. Corresponding parts of congruent triangles are congruent; therefore, $\overline{WT}$ and $\overline{WZ}$ are congruent, so we can set them equal to one another to find $x$: $2x = 3x - 5$ Subtract $2x$ from each side of the equation to isolate the variable on the left side of the equation: $0 = x - 5$ Add $5$ to each side to solve for $x$: $x = 5$ Now that we have the value for $x$, we can substitute it into the expression for $TW$: $TW = 2x$ Substitute $5$ for $x$: $TW = 2(5)$ Solve by multiplying: $TW = 10$