Question

uv ⊥ tu and qr ⊥ rs. complete the proof that ∠qrs ≅ ∠tuv.
statement\treason
1\tuv ⊥ tu\tgiven
2\tqr ⊥ rs\tgiven
3\tm∠tuv = 90°\t
4\tm∠qrs = 90°\t
5\tm∠tuv = m∠qrs\ttransitive property of equality
6\t∠qrs ≅ ∠tuv\t

Explanation:

Step1: Define right - angle property

If two lines are perpendicular, the angle formed between them is 90°. So, since $\overleftrightarrow{UV}\perp\overleftrightarrow{TU}$, then $m\angle TUV = 90^{\circ}$ (Definition of perpendicular lines).

Step2: Apply same property

Since $\overleftrightarrow{QR}\perp\overleftrightarrow{RS}$, then $m\angle QRS=90^{\circ}$ (Definition of perpendicular lines).

Step3: Use transitive property

We know that $m\angle TUV = 90^{\circ}$ and $m\angle QRS = 90^{\circ}$, so by the transitive property of equality ($a = c$ and $b = c$ implies $a = b$), $m\angle TUV=m\angle QRS$.

Step4: Recall angle - congruence definition

If the measures of two angles are equal, then the two angles are congruent. So, since $m\angle TUV=m\angle QRS$, $\angle QRS\cong\angle TUV$ (Definition of congruent angles).

Answer:

1. Definition of perpendicular lines
2. Definition of perpendicular lines
3. Transitive property of equality
4. Definition of congruent angles