Question

uv ⊥ vw and yz ⊥ xy. complete the proof that ∠uvw ≅ ∠xyz.
statement\treason
1 uv ⊥ vw\tgiven
2 yz ⊥ xy\tgiven
3 m∠uvw = 90°\t
4 m∠xyz = 90°\tangles forming a linear pair sum to 180°
5 m∠uvw = m∠xyz\tdefinition of angle bisector
6 ∠uvw ≅ ∠xyz\tdefinition of complementary angles
\tdefinition of perpendicular angles
\tdefinition of perpendicular lines
\tdefinition of supplementary angles

Explanation:

Step1: Recall perpendicular - angle relationship

If two lines are perpendicular, the angle formed by them is a right - angle. Since $\overleftrightarrow{UV}\perp\overleftrightarrow{VW}$, by the definition of perpendicular lines, the measure of $\angle UVW$ is $90^{\circ}$.

Step2: Recall perpendicular - angle relationship for the second pair

Since $\overleftrightarrow{YZ}\perp\overleftrightarrow{XY}$, by the definition of perpendicular lines, the measure of $\angle XYZ$ is $90^{\circ}$.

Step3: Equate the angle measures

If $m\angle UVW = 90^{\circ}$ and $m\angle XYZ=90^{\circ}$, then $m\angle UVW=m\angle XYZ$ by the transitive property of equality.

Step4: Use the definition of congruent angles

If two angles have the same measure, then they are congruent. So, $\angle UVW\cong\angle XYZ$ by the definition of congruent angles.

The reason for statement 3 is "Definition of perpendicular lines".
The reason for statement 4 is "Definition of perpendicular lines".
The reason for statement 5 is "Transitive property of equality" (not among the given options, but the correct logical reason for equating the two $90^{\circ}$ angle measures).
The reason for statement 6 is "Definition of congruent angles" (not among the given options in the dropdown, but the correct reason for concluding angle - congruence from equal angle measures).

Answer:

The reason for statement 3: Definition of perpendicular lines
The reason for statement 4: Definition of perpendicular lines