Question

\\( overleftrightarrow{wv} perp overleftrightarrow{uv} \\) and \\( overleftrightarrow{yz} perp overleftrightarrow{xy} \\). complete the proof that \\( angle xyzcongangle uvw \\).
statement\treason

1. \\( overleftrightarrow{wv} perp overleftrightarrow{uv} \\)\tgiven
2. \\( overleftrightarrow{yz} perp overleftrightarrow{xy} \\)\tgiven
3. \\( mangle uvw = 90^{circ} \\)\t
4. \\( mangle xyz = 90^{circ} \\)\tdefinition of perpendicular lines
5. \\( mangle uvw = mangle xyz \\)\ttransitive property of equality
6. \\( angle xyzcongangle uvw \\)\tdefinition of congruence

Explanation:

Step1: Recall perpendicular - lines property

If two lines are perpendicular, the angle formed by them is 90°. Since $\overleftrightarrow{WV}\perp\overleftrightarrow{UV}$, by the definition of perpendicular lines, $m\angle U VW = 90^{\circ}$.

Step2: Apply the same property to the other pair

Given $\overleftrightarrow{YZ}\perp\overleftrightarrow{XY}$, by the definition of perpendicular lines, $m\angle XYZ=90^{\circ}$.

Step3: Use the transitive property

Since $m\angle U VW = 90^{\circ}$ and $m\angle XYZ = 90^{\circ}$, by the transitive property of equality, $m\angle U VW=m\angle XYZ$.

Step4: Use the definition of congruent angles

If the measures of two angles are equal, then the angles are congruent. So, $\angle XYZ\cong\angle U VW$.

Answer:

The reason for statement 3 is "Definition of perpendicular lines".