1. The problem asks to determine if the statement "Two segments that have the same length must be congruent" is sometimes, always, or never true. 2. The definition of congruent segments states that two segments are congruent if and only if they have the same length. 3. Since the statement says the segments have the same length, by definition, they must be congruent. 4. Therefore, the statement is always true. 1. The problem asks if "If M is between A and B, then M bisects $\overline{AB}$" is sometimes, always, or never true. 2. A point bisects a segment if it divides the segment into two equal parts. 3. M being between A and B means $AM + MB = AB$, but it does not guarantee $AM = MB$. 4. Therefore, M does not necessarily bisect $\overline{AB}$. 5. The statement is sometimes true (if M is the midpoint) but not always. 1. The problem asks if "If Y is between X and Z, then X, Y, and Z are collinear" is sometimes, always, or never true. 2. By definition, if a point is between two others, all three points lie on the same line. 3. Therefore, the statement is always true. 1. Given $RS = 7y - 4$, $ST = y + 5$, and $RT = 28$, find $y$. 2. By the Segment Addition Postulate, $RS + ST = RT$. 3. Substitute values: $7y - 4 + y + 5 = 28$. 4. Simplify: $8y + 1 = 28$. 5. Subtract 1 from both sides: $8y = 27$. 6. Divide both sides by 8: $$y = \frac{27}{8}$$. 1. Given $RS = 2z + 6$, $ST = 4z - 3$, and $RT = 5z + 12$, find $z$. 2. By the Segment Addition Postulate, $RS + ST = RT$. 3. Substitute values: $2z + 6 + 4z - 3 = 5z + 12$. 4. Simplify left side: $6z + 3 = 5z + 12$. 5. Subtract $5z$ from both sides: $z + 3 = 12$. 6. Subtract 3 from both sides: $z = 9$. 1. Given $PT = 5x + 3$ and $TQ = 7x - 9$, find $PT$ if $PQ$ is the whole segment. 2. Since T is between P and Q, $PT + TQ = PQ$. 3. Without $PQ$ value, assume $PQ = PT + TQ$. 4. To find $PT$, set $PT = TQ$ for midpoint (if needed), or solve for $x$ if more info is given. 5. Since no total length is given, we cannot find numeric $PT$ without more info. 1. Given $PT = 4x - 6$ and $TQ = 3x + 4$, find $PT$. 2. Same as above, without total length, cannot find numeric $PT$. 1. Evaluate $|20 - 8|$. 2. Calculate inside absolute value: $20 - 8 = 12$. 3. Absolute value of 12 is 12. 1. Evaluate $|9 - 18|$. 2. Calculate inside absolute value: $9 - 18 = -9$. 3. Absolute value of -9 is 9. 1. Evaluate $-|4 - 27|$. 2. Calculate inside absolute value: $4 - 27 = -23$. 3. Absolute value of -23 is 23. 4. Apply negative sign: $-23$. 1. Simplify $8a - 3(4 + a) - 10$. 2. Distribute: $8a - 12 - 3a - 10$. 3. Combine like terms: $8a - 3a = 5a$, and $-12 - 10 = -22$. 4. Final expression: $5a - 22$. 1. Simplify $x + 2(5 - 2x) - (4 + 5x)$. 2. Distribute: $x + 10 - 4x - 4 - 5x$. 3. Combine like terms: $x - 4x - 5x = -8x$, and $10 - 4 = 6$. 4. Final expression: $-8x + 6$.