Question

complete the proof that (overleftrightarrow{uw} perp overleftrightarrow{vx}).
diagram: lines ( overleftrightarrow{uw} ) (horizontal, w left, u right) and ( overleftrightarrow{vx} ) (vertical, x top, v bottom) intersect at y, with a marked angle at y.

statement	reason
2. ( mangle uyv + mangle vyw = 180^circ )	angles forming a linear pair sum to ( 180^circ )
3. ( mangle uyv + mangle uyv = 180^circ )	substitution
4. ( mangle uyv = 90^circ )	blank
5. ( overleftrightarrow{uw} perp overleftrightarrow{vx} )	blank

Explanation:

Step1: Analyze Step 3

From step 3, we have \( m\angle UYV + m\angle UYV = 180^\circ \) (since \( \angle VYW\cong\angle UYV \), so \( m\angle VYW = m\angle UYV \)). This simplifies to \( 2m\angle UYV = 180^\circ \).

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

Divide both sides of \( 2m\angle UYV = 180^\circ \) by 2: \( m\angle UYV=\frac{180^\circ}{2} = 90^\circ \). The reason here is "Solving the equation \( 2m\angle UYV = 180^\circ \) for \( m\angle UYV \)" (or more precisely, "If \( 2x = 180^\circ \), then \( x = 90^\circ \) by division property of equality").

Step3: Reason for Step 5

If the measure of an angle between two lines (here, \( \angle UYV \) between \( \overleftrightarrow{UW} \) and \( \overleftrightarrow{VX} \)) is \( 90^\circ \), then the lines are perpendicular. So the reason for \( \overleftrightarrow{UW}\perp\overleftrightarrow{VX} \) is "If two lines intersect to form a right angle (90° angle), then the lines are perpendicular".

For Step 4, the reason is: "Division Property of Equality (from \( 2m\angle UYV = 180^\circ \), divide both sides by 2)".

For Step 5, the reason is: "Definition of Perpendicular Lines (if the angle between two lines is \( 90^\circ \), the lines are perpendicular)".

Answer:

• Step 4 Reason: Division Property of Equality (solving \( 2m\angle UYV = 180^\circ \) gives \( m\angle UYV = 90^\circ \))
• Step 5 Reason: Definition of Perpendicular Lines (if two lines form a \( 90^\circ \) angle, they are perpendicular)