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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

what answer should fill in the blank?

3 answers

The missing reason to fill in the blank is "substitution."

Thus, the complete proof segment would read as follows:

  1. ST¯¯¯¯¯¯¯ is the midsegment of trapezoid WXYZ.
  2. Given SU¯¯¯¯¯¯¯∥WX¯¯¯¯¯¯¯¯¯¯ UT¯¯¯¯¯¯¯∥ZY¯¯¯¯¯¯¯.
  3. Definition of midsegment (trapezoid).
  4. SU¯¯¯¯¯¯¯ is the midsegment of △WXZ.
  5. UT¯¯¯¯¯¯¯ is the midsegment of △XYZ.
  6. Definition of midsegment (triangle).
  7. SU = 1/2 WX
  8. UT = 1/2 ZY
  9. Midsegment Theorem.
  10. ST = SU + UT
  11. ST = 1/2 WX + 1/2 ZY (substitution)
  12. ST = 1/2 (WX + ZY) (Distributive Property)

In this sequence, the step that justifies the equality ST = 1/2 WX + 1/2 ZY is due to substitution.

The responses given are, "Midsegment Theorem, Segment Addition Postulate, Segment Subtraction Postulate, or Triangle Addition Postulate."

In this context, the correct reason needed to fill in the blank is "Midsegment Theorem."

So the proof would look like this:

  1. ST¯¯¯¯¯¯¯ is the midsegment of trapezoid WXYZ.
  2. Given SU¯¯¯¯¯¯¯∥WX¯¯¯¯¯¯¯¯¯¯ UT¯¯¯¯¯¯¯∥ZY¯¯¯¯¯¯¯.
  3. Definition of midsegment (trapezoid).
  4. SU¯¯¯¯¯¯¯ is the midsegment of △WXZ.
  5. UT¯¯¯¯¯¯¯ is the midsegment of △XYZ.
  6. Definition of midsegment (triangle).
  7. SU = 1/2 WX
  8. UT = 1/2 ZY
  9. Midsegment Theorem.
  10. ST = SU + UT
  11. ST = 1/2 WX + 1/2 ZY (substitution)
  12. ST = 1/2 (WX + ZY) (Distributive Property)

Since the Midsegment Theorem states that the midsegment is parallel to the bases and its length is related to the lengths of the bases, that would be the appropriate justification for step 9.