Question

rs ⊥ qr and uv ⊥ tu. complete the proof that ∠tuv ≅ ∠qrs.
statement reason
1 rs ⊥ qr given
2 uv ⊥ tu given
3 m∠qrs = 90°
4 m∠tuv = 90° definition of perpendicular lines
5 m∠qrs = m∠tuv transitive property of equality
6 ∠tuv ≅ ∠qrs definition of congruence

Explanation:

Step1: Recall perpendicular - lines property

The definition of perpendicular lines states that if two lines are perpendicular, the angle formed between them is a right - angle, which measures 90°. Since $\overrightarrow{RS}\perp\overrightarrow{QR}$, by the definition of perpendicular lines, the measure of $\angle QRS$ is 90°.

Step2: Recall perpendicular - lines property for second pair

Similarly, for $\overrightarrow{UV}\perp\overrightarrow{TU}$, by the definition of perpendicular lines, the measure of $\angle TUV$ is 90°.

Step3: Apply transitive property

We know that $m\angle QRS = 90^{\circ}$ and $m\angle TUV=90^{\circ}$. By the transitive property of equality (if $a = c$ and $b = c$, then $a = b$), we have $m\angle QRS=m\angle TUV$.

Step4: Use congruence definition

Since the measures of $\angle QRS$ and $\angle TUV$ are equal, by the definition of congruent angles (two angles are congruent if and only if their measures are equal), $\angle TUV\cong\angle QRS$.

Answer:

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