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 midpoint postulates: segment addition postulate

Explanation:

Step1: Identify property for statement 1

If a quantity is equal to itself, and we go from equality to congruence of segments, it's the definition of congruence.
Definition of Congruence

Step2: Identify property for statement 2

If a point is between two others, the sum of the two - smaller segments equals the larger one, which is the Segment Addition Postulate.
Segment Addition Postulate

Step3: Identify property for statement 3

A segment is congruent to itself, which is the Reflexive Property of Congruence.
Reflexive Property of Congruence

Step4: Identify property for statement 4

Adding the same quantity to both sides of an equation, it's the Addition Property of Equality.
Addition Property of Equality

Step5: Identify property for statement 5

If \(AB = DE\), then \(DE=AB\), which is the Symmetric Property of Equality.
Symmetric Property of Equality

Step6: Identify property for statement 6

If a point is a mid - point of a segment, the two sub - segments are equal, which is the Definition of Midpoint.
Definition of Midpoint

Step7: Identify property for statement 7

If two segments are congruent to a third segment, then they are congruent to each other, which is the Transitive Property of Congruence.
Transitive Property of Congruence

Step8: Identify property for statement 8

Subtracting the same quantity from both sides of an equation, it's the Subtraction Property of Equality.
Subtraction Property of Equality

Step9: Identify property for statement 9

Replacing \(RS\) with \(WX\) since \(RS = WX\), it's the Substitution Property of Equality.
Substitution Property of Equality

Step10: Identify property for statement 10

If two equal segments are on the same line and one point is between the other two, the middle point is the mid - point, which is the Definition of Midpoint.
Definition of Midpoint

Step11: Identify property for statement 11

If a segment is congruent to itself, and we go from congruence to equality, it's the Definition of Congruence.
Definition of Congruence

Step12: Identify property for statement 12

Isolating a term in an addition equation, it's the Subtraction Property of Equality.
Subtraction Property of Equality

Step13: Identify property for statement 13

Dividing both sides of an equation by 2, it's the Division Property of Equality.
Division Property of Equality

Step14: Identify property for statement 14

If \(RS = ST\) and \(ST = 2UV\), then \(RS = 2UV\), which is the Transitive Property of Equality.
Transitive Property of Equality

Answer:

1. Definition of Congruence
2. Segment Addition Postulate
3. Reflexive Property of Congruence
4. Addition Property of Equality
5. Symmetric Property of Equality
6. Definition of Midpoint
7. Transitive Property of Congruence
8. Subtraction Property of Equality
9. Substitution Property of Equality
10. Definition of Midpoint
11. Definition of Congruence
12. Subtraction Property of Equality
13. Division Property of Equality
14. Transitive Property of Equality