Question

$\angle swt \cong \angle twu$. complete the proof that $\overleftrightarrow{tv} \perp \overleftrightarrow{su}$.

diagram: two lines intersecting at $w$, labeled $s, v$ on one line, $t, u$ on the other

statement	reason
2. $m\angle swt + m\angle twu = 180^\circ$	angles forming a linear pair sum to $180^\circ$
3. $m\angle swt + m\angle swt = 180^\circ$	substitution
4. $m\angle swt = 90^\circ$	properties of addition, subtraction, multiplication, and division
5. $\overleftrightarrow{tv} \perp \overleftrightarrow{su}$	reason to be added

Explanation:

Step1: Recall the definition of perpendicular lines

Two lines are perpendicular if the angle between them is \(90^\circ\). In this case, we need to show that the angle between \(\overleftrightarrow{TV}\) and \(\overleftrightarrow{SU}\) (which is \(\angle SWT\)) is \(90^\circ\) to conclude they are perpendicular.

Step2: Use the result from step 4

From step 4, we know that \(m\angle SWT = 90^\circ\). By the definition of perpendicular lines (if the measure of the angle between two intersecting lines is \(90^\circ\), then the lines are perpendicular), we can conclude that \(\overleftrightarrow{TV}\perp\overleftrightarrow{SU}\). The reason for statement 5 is "Definition of perpendicular lines (if two lines intersect to form a \(90^\circ\) angle, then they are perpendicular)".

Answer:

The reason for statement 5 (\(\overleftrightarrow{TV}\perp\overleftrightarrow{SU}\)) is "Definition of perpendicular lines (if two lines intersect to form a \(90^\circ\) angle, then the lines are perpendicular)".