Question

lesson 6 - 2

1. complete the proof.

given: ( xy = 6 ), ( xz = 14 )
prove: ( yz = 8 )

statements	reasons
2. ( xy+yz = xz )	2.
3. ( 6 + yz = 14 )	3.
4. ( yz = 8 )	4.

1. based on the given information in item 9, can you conclude that ( y ) is the midpoint of ( xz )? explain your reasoning.

Explanation:

Problem 9:

Step1: Identify Reason for Statement 1

The first statement is just restating the given information, so the reason is "Given".

Step2: Identify Reason for Statement 2

The equation \(XY + YZ=XZ\) comes from the Segment Addition Postulate, which states that if a point \(Y\) is on segment \(XZ\), then \(XY + YZ = XZ\). So the reason is "Segment Addition Postulate".

Step3: Identify Reason for Statement 3

We substitute \(XY = 6\) (from Statement 1) into the equation from Statement 2, so the reason is "Substitution Property of Equality".

Step4: Identify Reason for Statement 4

To solve \(6+YZ = 14\) for \(YZ\), we subtract 6 from both sides (Subtraction Property of Equality), so \(YZ=14 - 6=8\). The reason is "Subtraction Property of Equality".

A midpoint of a segment divides the segment into two equal - length segments. For \(Y\) to be the midpoint of \(XZ\), we need \(XY=YZ\). We know that \(XY = 6\) and \(YZ = 8\). Since \(6
eq8\), \(Y\) does not divide \(XZ\) into two equal parts.

Answer:

Statements	Reasons
2. \(XY+YZ = XZ\)	2. Segment Addition Postulate
3. \(6 + YZ=14\)	3. Substitution Property of Equality
4. \(YZ = 8\)	4. Subtraction Property of Equality

Problem 10: