Question

pq ⊥ qr and tu ⊥ st. complete the proof that ∠pqr ≅ ∠stu.
statement\treason
1\tpq ⊥ qr\tgiven
2\ttu ⊥ st\tgiven
3\tm∠pqr = 90°\tgiven
4\tm∠stu = 90°\tdefinition of perpendicular lines
5\tm∠pqr = m∠stu\tdefinition of perpendicular lines
6\t∠pqr ≅ ∠stu

Explanation:

Step1: Recall the given information

$\overrightarrow{PQ}\perp\overrightarrow{QR}$ and $\overrightarrow{TU}\perp\overrightarrow{ST}$ are given.

Step2: Use the definition of perpendicular lines

By the definition of perpendicular lines, if two lines are perpendicular, the angle formed between them is $90^{\circ}$. So $m\angle PQR = 90^{\circ}$ and $m\angle STU=90^{\circ}$ (Statements 3 and 4).

Step3: Establish equality of angle - measures

Since $m\angle PQR = 90^{\circ}$ and $m\angle STU = 90^{\circ}$, we have $m\angle PQR=m\angle STU$ (Statement 5).

Step4: Recall the definition of congruent angles

Two angles are congruent if and only if their measures are equal. Since $m\angle PQR=m\angle STU$, then $\angle PQR\cong\angle STU$ (Statement 6). The reason for Statement 6 is "If two angles have the same measure, then they are congruent" or "Definition of congruent angles".

Answer:

The reason for statement 6 is "Definition of congruent angles".