Question

∠vwz≅∠wzx. complete the proof that (overleftrightarrow{vx}perpoverleftrightarrow{wy}).
statement reason
1 ∠vwz≅∠wzx given
2 (mangle vwz + mangle wzx=180^{circ}) angles forming a linear pair sum to (180^{circ})
3 (mangle vwz + mangle vwz = 180^{circ}) substitution
4 (mangle vwz=90^{circ}) properties of addition, subtraction, multiplication, and division
5 (overleftrightarrow{vx}perpoverleftrightarrow{wy})

Explanation:

Step1: Recall perpendicular - line definition

Two lines are perpendicular if the angle between them is 90°.

Step2: Analyze given information

We are given that $\angle VZW\cong\angle WZX$ and $\angle VZW$ and $\angle WZX$ are a linear - pair (since they are adjacent and their non - common sides form a straight line).

Step3: Use linear - pair property

Since angles forming a linear pair sum to 180°, and $\angle VZW$ and $\angle WZX$ are a linear pair, we have $m\angle VZW + m\angle WZX=180^{\circ}$.

Step4: Apply substitution

Because $\angle VZW\cong\angle WZX$, we can substitute $m\angle WZX$ with $m\angle VZW$ in the equation $m\angle VZW + m\angle WZX = 180^{\circ}$, getting $m\angle VZW+m\angle VZW = 180^{\circ}$, or $2m\angle VZW=180^{\circ}$.

Step5: Solve for the angle measure

Dividing both sides of the equation $2m\angle VZW = 180^{\circ}$ by 2 using the division property of equality, we find $m\angle VZW = 90^{\circ}$.

Step6: Determine perpendicularity

Since the angle between $\overrightarrow{VX}$ and $\overrightarrow{WY}$ at point $Z$ (i.e., $\angle VZW$) is 90°, by the definition of perpendicular lines (if the measure of the angle between two lines is 90°, the lines are perpendicular), we can conclude that $\overrightarrow{VX}\perp\overrightarrow{WY}$. The reason for step 5 is "Definition of perpendicular lines (if an angle between two lines is 90°, the lines are perpendicular)".

Answer:

Definition of perpendicular lines (if an angle between two lines is 90°, the lines are perpendicular)