Question

complete the proof that ( overleftrightarrow{su} perp overleftrightarrow{tv} ).

Explanation:

Step1: Substitute angles

Since \( \angle SWT\cong\angle TWU\), we can substitute \(m\angle TWU\) with \(m\angle SWT\) in the equation \(m\angle SWT + m\angle TWU=180^{\circ}\). So, \(m\angle SWT + m\angle SWT = 180^{\circ}\) (Substitution Property of Equality).

Step2: Solve for \(m\angle SWT\)

Combine like - terms: \(2m\angle SWT=180^{\circ}\), then divide both sides by 2: \(m\angle SWT = 90^{\circ}\).

Step3: Use the definition of perpendicular lines

If the measure of the angle between two lines is \(90^{\circ}\), then the lines are perpendicular. Since \(m\angle SWT = 90^{\circ}\), by the Definition of perpendicular lines, \(\overleftrightarrow{SU}\perp\overleftrightarrow{TV}\).

Answer:

1. Given
2. Definition of a linear pair (not perpendicular lines, the correct reason for \(m\angle SWT + m\angle TWU = 180^{\circ}\) is that \(\angle SWT\) and \(\angle TWU\) form a linear pair)
3. Substitution Property of Equality
4. Properties of addition, subtraction, multiplication, and division (simplify \(2m\angle SWT=180^{\circ}\) to \(m\angle SWT = 90^{\circ}\))
5. Definition of perpendicular lines (since \(m\angle SWT = 90^{\circ}\), \(\overleftrightarrow{SU}\perp\overleftrightarrow{TV}\))