Question

proof write a paragraph proof to prove that if ( pq = 4(x - 3)+1 ), ( qr=x + 10 ), and ( x = 7 ), then ( overline{pq}congoverline{qr} ).
given: ( pq = 4(x - 3)+1 ), ( qr=x + 10 ), and ( x = 7 )
prove: ( overline{pq}congoverline{qr} )
proof: from the given, ( pq = 4(x - 3)+1 ) and ( qr=x + 10 ). because ( x = 7 ), ( pq=) select choice and ( qr=) select choice by the select choice. by select choice, ( pq = qr ). any two line segments that have the same length select choice congruent, so ( overline{pq}congoverline{qr} ).

Explanation:

Step1: Substitute \(x = 7\) into \(PQ\)

Substitute \(x = 7\) into \(PQ=4(x - 3)+1\).

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Step2: Substitute \(x = 7\) into \(QR\)

Substitute \(x = 7\) into \(QR=x + 10\).

$$QR=7+10=17$$

Step3: Compare \(PQ\) and \(QR\)

Since \(PQ = 17\) and \(QR=17\), by the substitution property (substituting the value of \(x\) into the expressions for \(PQ\) and \(QR\)), \(PQ=QR\). By the definition of congruent segments (if two segments have the same length, they are congruent), any two line segments that have the same length are congruent.

Answer:

From the given, \(PQ = 4(x - 3)+1\) and \(QR=x + 10\). Because \(x = 7\), \(PQ=4(7 - 3)+1=17\) and \(QR=7 + 10=17\) by the substitution property. By substitution (equality of their lengths), \(PQ = QR\). Any two line segments that have the same length are congruent, so \(\overline{PQ}\cong\overline{QR}\).