Question

complete the proof that $overleftrightarrow{uw}perpoverleftrightarrow{vx}$. 1. $angle vywcongangle uyv$ given 2. $mangle uyv + mangle vyw=180^{circ}$ angles forming a linear pair sum to $180^{circ}$ 3. $mangle uyv + mangle uyv=180^{circ}$ 4. $mangle uyv = 90^{circ}$ properties of addition, subtraction, multiplication, and division 5. $overleftrightarrow{uw}perpoverleftrightarrow{vx}$ definition of perpendicular lines

Explanation:

Step1: Substitute equal - angle measures

Since $\angle VYW\cong\angle UYV$ (given), we can substitute $m\angle VYW$ with $m\angle UYV$ in the equation $m\angle UYV + m\angle VYW=180^{\circ}$ (from step 2). So we get $m\angle UYV + m\angle UYV = 180^{\circ}$.

Step2: Solve for $m\angle UYV$

Let $x = m\angle UYV$. The equation $x + x=180^{\circ}$ simplifies to $2x = 180^{\circ}$. Dividing both sides by 2 gives $x = 90^{\circ}$, so $m\angle UYV=90^{\circ}$.

Step3: Use perpendicular - line definition

Since $\angle UYV$ is the angle formed by $\overleftrightarrow{UW}$ and $\overleftrightarrow{VX}$, and $m\angle UYV = 90^{\circ}$, by the definition of perpendicular lines (two lines are perpendicular if the angle between them is $90^{\circ}$), we have $\overleftrightarrow{UW}\perp\overleftrightarrow{VX}$.

Answer:

The reason for step 3 is "Substitution property of equality".