Question

complete the proof by choosing the correct
eason\.

given: \\(qs = tu\\)
\\(rs = tr\\)

prove: \\(qr = ru\\)

\

$$\begin{array}{|l|l|} \\hline \\text{statement} & \\text{reason} \\\\ \\hline 1. \\ qs = qr + rs & \\text{reason ?} \\\\ \\hline 2. \\ tu = tr + ru & \\text{reason ?} \\\\ \\hline 3. \\ qs = tu & \\text{reason ?} \\\\ \\hline 4. \\ qr + rs = tr + ru & \\text{reason ?} \\\\ \\hline 5. \\ rs = tr & \\text{reason ?} \\\\ \\hline 6. \\ qr + tr = tr + ru & \\text{reason ?} \\\\ \\hline 7. \\ qr = ru & \\text{reason ?} \\\\ \\hline \\end{array}$$

Explanation:

⚡ Using what you learned: proofs involving angles, segments, triangles, and parallelograms

Step 1: Analyze Statements 1 and 2

The statements are:

1. \( QS = QR + RS \)
2. \( TU = TR + RU \)

Both statements express that a whole segment is equal to the sum of its non-overlapping parts.

• Reason 1 & 2: Segment Addition Postulate

Step 2: Analyze Statement 3

The statement is:

1. \( QS = TU \)

This is given information in the problem description.

• Reason 3: Given

Step 3: Analyze Statement 4

The statement is:

1. \( QR + RS = TR + RU \)

This is obtained by substituting the expressions from Statements 1 and 2 into Statement 3.

• Reason 4: Substitution Property of Equality

Step 4: Analyze Statement 5

The statement is:

1. \( RS = TR \)

This is given information in the problem description.

• Reason 5: Given

Step 5: Analyze Statement 6

The statement is:

1. \( QR + TR = TR + RU \)

This is obtained by substituting \( TR \) for \( RS \) (from Statement 5) into Statement 4.

• Reason 6: Substitution Property of Equality

Step 7: Analyze Statement 7

The statement is:

1. \( QR = RU \)

This is obtained by subtracting \( TR \) from both sides of Statement 6.

• Reason 7: Subtraction Property of Equality

Answer:

The complete proof with reasons is:

Statement	Reason
2. \( TU = TR + RU \)	Segment Addition Postulate
3. \( QS = TU \)	Given
4. \( QR + RS = TR + RU \)	Substitution Property of Equality
5. \( RS = TR \)	Given
6. \( QR + TR = TR + RU \)	Substitution Property of Equality
7. \( QR = RU \)	Subtraction Property of Equality