Answer: To prove the statements, let's go step by step:
1. Given that Y is the midpoint of line XZ.
2. Given that line XY = line XZ.
- The reason for this is that Y is the midpoint of line XZ, so line XY and line YZ are equal in length.
3. Given that line XY + line XY = line XY + line YZ.
- The reason for this is the reflexive property of equality. When you add the same segment to both sides of an equation, the equality remains true.
4. Given that line XYZ = line XY + line YZ.
- The reason for this is the definition of the line segment addition postulate. It states that the length of a line segment is equal to the sum of the lengths of its parts.
5. Given that line XY + line YZ = line XZ.
- The reason for this is the substitution property of equality. Since line XY is equal to line XZ, we can substitute line XY + line YZ with line XZ.
6. Given that line XYZ = line XZ.
- The reason for this is the transitive property of equality. If line XYZ is equal to line XY + line YZ, and line XY + line YZ is equal to line XZ, then line XYZ is also equal to line XZ.
7. Given that line XY = 1/2 of line XZ.
- The reason for this is the definition of a midpoint. A midpoint divides a line segment into two equal parts. Since Y is the midpoint of line XZ, line XY is half the length of line XZ.
By following these logical steps and properties of equality, we have proven the given statements.
Explanation: