The proof for the Trapezoid Midsegment Theorem is partially filled in. Supply the missing reason to prove that the length of the midsegment is half the sum of the lengths of the bases. (1 point) ST¯¯¯¯¯¯¯ is the midsegment of trapezoid WXYZ . given SU¯¯¯¯¯¯¯∥WX¯¯¯¯¯¯¯¯¯¯ UT¯¯¯¯¯¯¯∥ZY¯¯¯¯¯¯¯ definition of midsegment (trapezoid) SU¯¯¯¯¯¯¯ is the midsegment of △WXZ . UT¯¯¯¯¯¯¯ is the midsegment of △XYZ . definition of midsegment (triangle) SU=12WX UT=12ZY Midsegment Theorem ST=SU+UT ST=12WX+12ZY substitution ST=12(WX+ZY) Distributive Property

Answer 1

To complete the proof, the missing reason for the line "ST = SU + UT" can be provided as follows:

Reason: The lengths of segments addition postulate (segments can be added together).

Here’s how the reasoning would fit into the proof:

1. ST is the midsegment of trapezoid WXYZ, given \(SU \parallel WX\) and \(UT \parallel ZY\) (definition of midsegment).
2. \(SU\) is the midsegment of triangle \(WXZ\) (definition of midsegment).
3. \(UT\) is the midsegment of triangle \(XYZ\) (definition of midsegment).
4. \(SU = \frac{1}{2} WX\) (Midsegment Theorem).
5. \(UT = \frac{1}{2} ZY\) (Midsegment Theorem).
6. ST = SU + UT (Lengths of segments addition postulate).
7. \(ST = \frac{1}{2} WX + \frac{1}{2} ZY\) (substitution).
8. \(ST = \frac{1}{2}(WX + ZY)\) (Distributive Property).

This provides a complete and logical proof of the Trapezoid Midsegment Theorem.