Question

practice! justify each statement below using a property of equality, property of congruence, definition, or postulate.

1. if pq = pq then $overline{pq}congoverline{pq}$ 1.
2. if k is between j and l then jk + kl = jl 2.
3. $overline{ef}congoverline{ef}$ 3.
4. if rs = tu then rs + xy = tu + xy 4.
5. if ab = de then de = ab 5.
6. if y is the mid - point of $overline{xz}$, then xy = yz 6.
7. if $overline{fg}congoverline{hi}$ and $overline{hi}congoverline{jk}$, then $overline{fg}congoverline{jk}$ 7.
8. if ab + cd = ef + cd, then ab = ef 8.
9. if pq + rs = tv and rs = wx, then pq + wx = tv 9.
10. if lp = pn, and l, p, and n are collinear, then p is the mid - point of $overline{ln}$ 10.
11. if $overline{uv}congoverline{uv}$, then uv = uv 11.
12. if cd + de = ce, then cd = ce - de 12.
13. if 2xy = xz then xy=$\frac{1}{2}$xz 13.
14. if rs = st and st = 2uv, then rs = 2uv 14.

reasons bank
properties of equality: addition property, subtraction property, multiplication property, division property, distributive property, substitution property, reflexive property, symmetric property, transitive property
properties of congruence: reflexive property, symmetric property, transitive property
definitions: definition of congruence, definition of mid - point
postulates: segment addition postulate

Explanation:

Step1: Identify property for "If PQ = PQ then $\overline{PQ}\cong\overline{PQ}$"

By reflexive property of equality, a quantity is equal to itself, and for congruence, a segment is congruent to itself. So it's reflexive property of congruence.

Step2: Identify property for "If K is between J and L then JK + KL=JL"

This is the segment - addition postulate which states that if a point K lies on a line segment $\overline{JL}$, then the sum of the lengths of the two sub - segments $\overline{JK}$ and $\overline{KL}$ is equal to the length of the entire segment $\overline{JL}$.

Step3: Identify property for "$\overline{EF}\cong\overline{EF}$"

It's the reflexive property of congruence as a segment is congruent to itself.

Step4: Identify property for "If RS = TU then RS+XY = TU + XY"

This is the addition property of equality. If two quantities are equal, adding the same quantity to both sides of the equation keeps the equation true.

Step5: Identify property for "If AB = DE then DE = AB"

This is the symmetric property of equality which states that if a = b, then b = a.

Step6: Identify property for "If Y is the mid - point of $\overline{XZ}$, then XY = YZ"

This is the definition of a mid - point which states that the mid - point of a line segment divides the segment into two equal parts.

Step7: Identify property for "If $\overline{FG}\cong\overline{HI}$ and $\overline{HI}\cong\overline{JK}$, then $\overline{FG}\cong\overline{JK}$"

This is the transitive property of congruence which states that if a $\cong$ b and b $\cong$ c, then a $\cong$ c.

Step8: Identify property for "If AB + CD=EF + CD then AB = EF"

This is the subtraction property of equality. If we subtract the same quantity (CD in this case) from both sides of the equation, the equality still holds.

Step9: Identify property for "If PQ + RS = TV and RS = WX, then PQ+WX = TV"

This is the substitution property of equality. Since RS and WX are equal, we can substitute WX for RS in the first equation.

Step10: Identify property for "If LP = PN, and L, P, and N are collinear, then P is the mid - point of $\overline{LN}$"

This is the definition of a mid - point. If a point P on a line segment $\overline{LN}$ makes LP = PN, then P is the mid - point of $\overline{LN}$.

Step11: Identify property for "If $\overline{UV}\cong\overline{UV}$, then UV = UV"

This is the definition of congruence. If two segments are congruent, then their lengths are equal.

Step12: Identify property for "If CD + DE = CE, then CD = CE - DE"

This is the subtraction property of equality. Subtracting DE from both sides of the equation CD + DE = CE gives CD = CE - DE.

Step13: Identify property for "If 2XY = XZ then XY=$\frac{1}{2}$XZ"

This is the division property of equality. Dividing both sides of the equation 2XY = XZ by 2 gives XY=$\frac{1}{2}$XZ.

Step14: Identify property for "If RS = ST and ST = 2UV, then RS = 2UV"

This is the transitive property of equality. If a = b and b = c, then a = c.

Answer:

1. Reflexive property of congruence
2. Segment - addition postulate
3. Reflexive property of congruence
4. Addition property of equality
5. Symmetric property of equality
6. Definition of mid - point
7. Transitive property of congruence
8. Subtraction property of equality
9. Substitution property of equality
10. Definition of mid - point
11. Definition of congruence
12. Subtraction property of equality
13. Division property of equality
14. Transitive property of equality