Question

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

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 midpoint
postulates:
segment addition postulate

Explanation:

Step1: Analyze statement 7

Given $\overline{FG}\cong\overline{HI}$ and $\overline{HI}\cong\overline{JK}$, by the Transitive Property of Congruence, if two segments are congruent to the same segment, they are congruent to each other. So the reason for statement 7 is Transitive Property of Congruence.

Step2: Analyze statement 8

Given $AB + CD=EF + CD$, subtracting $CD$ from both sides of the equation using the Subtraction Property of Equality gives $AB = EF$. So the reason for statement 8 is Subtraction Property of Equality.

Step3: Analyze statement 9

Given $PQ+RS = TV$ and $RS = WX$, substituting $WX$ for $RS$ in the first - equation $PQ+RS = TV$ gives $PQ + WX=TV$ by the Substitution Property of Equality. So the reason for statement 9 is Substitution Property of Equality.

Step4: Analyze statement 10

Given $LP = PN$ and $L$, $P$, and $N$ are collinear, by the Definition of Mid - point, $P$ is the midpoint of $\overline{LN}$. So the reason for statement 10 is Definition of Midpoint.

Step5: Analyze statement 11

Given $\overline{UV}\cong\overline{UV}$, by the Definition of Congruence (if two segments are congruent, their lengths are equal), $UV = UV$. So the reason for statement 11 is Definition of Congruence.

Step6: Analyze statement 12

Given $CD + DE=CE$, subtracting $DE$ from both sides of the equation using the Subtraction Property of Equality gives $CD=CE - DE$. So the reason for statement 12 is Subtraction Property of Equality.

Step7: Analyze statement 13

Given $2XY = XZ$, dividing both sides of the equation by 2 using the Division Property of Equality gives $XY=\frac{1}{2}XZ$. So the reason for statement 13 is Division Property of Equality.

Step8: Analyze statement 14

Given $RS = ST$ and $ST = 2UV$, by the Transitive Property of Equality, $RS = 2UV$. So the reason for statement 14 is Transitive Property of Equality.

Answer:

1. Transitive Property of Congruence
2. Subtraction Property of Equality
3. Substitution Property of Equality
4. Definition of Midpoint
5. Definition of Congruence
6. Subtraction Property of Equality
7. Division Property of Equality
8. Transitive Property of Equality