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Given: AB = 28; M lies on the line AB Prove: M is a midpoint. 1) M lies on AB AB = 28 _____ 2) AM + MB = AB _____ 3) 2x + 2 + 4x - 10 = 28 _____ 4) 6x - 8 = 28 _____ 5) 6x = 36 _____ 6) x = 6 _____ 7) AM = 2(6) + 2 and MB = 4(6) - 10 _____ 8) AM = 14 and MB = 14 _____ 9) M is a midpoint _____

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The proof you have presented demonstrates that point M is the midpoint of line segment AB, given that AB equals 28. Here’s the proof formatted more clearly and logically:

Given:
AB = 28
M lies on the line segment AB

Proof:

  1. M lies on AB.
    Statement: M is a point on the line segment AB. (This is given.)

  2. AM + MB = AB
    Reason: By the definition of a line segment, the length of AB is equal to the sum of the segments AM and MB, i.e., AM + MB = AB.

  3. Let AM = 2x + 2 and MB = 4x - 10.
    Statement: Assign expressions for AM and MB in terms of x (which will be solved later).

  4. Substituting these into the equation:
    We have AM + MB = AB → (2x + 2) + (4x - 10) = 28.

  5. Simplifying the equation:
    2x + 2 + 4x - 10 = 28
    Combine like terms: 6x - 8 = 28.

  6. Adding 8 to both sides:
    6x = 36.

  7. Dividing by 6:
    x = 6.

  8. Finding AM and MB:
    Now substituting x back into the expressions for AM and MB:
    AM = 2(6) + 2 = 12 + 2 = 14.
    MB = 4(6) - 10 = 24 - 10 = 14.

  9. Conclusion:
    Since AM = 14 and MB = 14, we can conclude that AM = MB.
    Therefore, M is the midpoint of line segment AB, as it divides AB into two equal parts.

Thus, we have proven that M is a midpoint of segment AB.

Options you can use to fill in the blanks are, addition property of equality, combine like terms, definition of midpoint, division property of equality, given, segment addition postulate, substitution property, substitution property, simplifying

Certainly! Here is the proof with the appropriate statements filled in using the given options:

Given:
AB = 28
M lies on the line segment AB

Proof:

  1. M lies on AB.
    Statement: Given.

  2. AM + MB = AB
    Statement: Segment Addition Postulate.

  3. 2x + 2 + 4x - 10 = 28
    Statement: Substitution Property. (Substituting expressions for AM and MB)

  4. 6x - 8 = 28
    Statement: Combine Like Terms.

  5. 6x = 36
    Statement: Addition Property of Equality. (Adding 8 to both sides)

  6. x = 6
    Statement: Division Property of Equality. (Dividing by 6)

  7. AM = 2(6) + 2 and MB = 4(6) - 10
    Statement: Substitution Property. (Substituting x back into the expressions)

  8. AM = 14 and MB = 14
    Statement: Simplifying. (Calculating the values of AM and MB)

  9. M is a midpoint
    Statement: Definition of Midpoint. (Since AM = MB, M is the midpoint of AB)

Thus, we have proven that M is the midpoint of segment AB using the appropriate mathematical properties.