Question

1. given: 2pq = pr

prove: q is the midpoint of \overline{pr}

statements	reasons
2. pr = pq + qr	2.
3. 2pq = pq + qr	3.
4. pq = qr	4.
5.	5. def. of ________

(diagram: a line segment with points p, q, r in sequence)

Explanation:

Step1: Fill Statement 1

The given is \( 2PQ = PR \), so statement 1 is \( 2PQ = PR \).

Step2: Reason for Statement 2

The formula \( PR = PQ + QR \) is the Segment Addition Postulate (which states that if a point \( Q \) is between \( P \) and \( R \), then \( PR = PQ + QR \)). So reason 2 is "Segment Addition Postulate".

Step3: Reason for Statement 3

We substitute \( PR \) with \( 2PQ \) (from statement 1) into the equation from statement 2. So reason 3 is "Substitution Property (substituting \( PR \) with \( 2PQ \))".

Step4: Reason for Statement 4

Subtract \( PQ \) from both sides of \( 2PQ = PQ + QR \). Using the Subtraction Property of Equality (if \( a = b + c \), then \( a - b = c \)), we get \( PQ = QR \). So reason 4 is "Subtraction Property of Equality".

Step5: Fill Statement 5 and Reason

Since \( PQ = QR \), by the definition of a midpoint (a midpoint of a segment divides the segment into two congruent segments), \( Q \) is the midpoint of \( \overline{PR} \). So statement 5 is " \( Q \) is the midpoint of \( \overline{PR} \)" and reason 5 is "Def. of midpoint".

Answer:

1. Statement 1: \( 2PQ = PR \)
2. Reason 2: Segment Addition Postulate
3. Reason 3: Substitution Property
4. Reason 4: Subtraction Property of Equality
5. Statement 5: \( Q \) is the midpoint of \( \overline{PR} \); Reason 5: Def. of midpoint