Question

1. complete the sentence in a two - column the right. monitoring progress and modelin in exercises 3 and 4, copy and complete the proof. 3. given pq = rs prove pr = qs 4. given ∠1 is a complement of ∠2. ∠2≅∠3 prove ∠1 is a complement of ∠3. in exercises 5 - 10, name the property that the state illustrates. (see example 2.) 5. if (overline{pq}congoverline{st}) and (overline{st}congoverline{uv}), then (overline{pq}congoverline{uv}). 6. ∠f≅∠f

Explanation:

Step1: Analyze the line - segment problem

Given \(PQ = RS\). By the segment - addition postulate, \(PR=PQ + QR\) and \(QS=QR + RS\).
Since \(PQ = RS\), we can substitute \(PQ\) for \(RS\) in the equation for \(QS\).

Step2: Substitute and simplify

\(QS=QR + PQ\). And \(PR = PQ+QR\). So, \(PR = QS\).

Step3: Analyze the angle - complement problem

Given \(\angle1\) is a complement of \(\angle2\), so \(\angle1+\angle2 = 90^{\circ}\). Also given \(\angle2\cong\angle3\), which means \(m\angle2=m\angle3\).

Step4: Substitute the angle measures

Substitute \(m\angle3\) for \(m\angle2\) in the equation \(\angle1+\angle2 = 90^{\circ}\). We get \(\angle1+\angle3 = 90^{\circ}\). So, \(\angle1\) is a complement of \(\angle3\).

Answer:

For problem 3: The proof is completed as shown above. For problem 4: The proof is completed as shown above. For problem 5: The property illustrated is the transitive property of congruence for segments. For problem 6: The property illustrated is the reflexive property of congruence for angles.