___(0-10 pts) Describe the segment properties of
congruence. Give at least 3 examples.
Symmetric Property of Congruence:
If segment AB is congruent to segment CD, then segment CD is congruent to segment AB. This is because both have the same measurements and can easily substitute each other.
(
= 
, 
= 
)
Examples:
EXAMPLE 1
Given : 
= 
Prove : 
= 
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Statements |
Reasons |
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1. |
1. Given |
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2. PQ = XY |
2. Definition of congruent segment |
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3. XY = PQ |
3. Symmetric property of equality |
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4. |
4. Definition of congruent segements |
Paragraph Proof: You are given that 
. By the definition of congruent segments, PQ = XY. By the symmetric property of equality, XY = PQ . Therefore, by the definition of congruent segments, if follows that 
.
EXAMPLE 2
More Examples on the Symmetric Property:
1)
Given: XY= 8, XZ=8, and the length of XY is congruent to the length of ZY(could not write it)
Prove: the length of XZ is congruent to the length of ZY
Statement Reasoning
XY=8, XZ=8 Given
X=XY
XY=XZ Transitive Property of Equality
The length of XY is congruent to the length of XZ Definition of Congruent Segments
The length of XY is congruent to the length of ZY Given
The length of XZ is congruent to the length of ZY Transitive Property of Segment Congruency
2)
Given: The length of VW is congruent to the length of WX, The length of WX is contruent to the length of YZ
Prove: The length of UW is congruent to the length of XZ
Statement Reasoning
The Length of UV is congruent to the length of XY,
The length of VW is congruent to the length of WX, Given
The length of WX is congruent to the length of YZ
THe length of VW is congruent to the length of YZ Definition of congruent segments
UV=XY, VW=YZ Definition of congruent segments
UV+VW=XY+YZ Segment Addition Postulate
UV+VW=UW, XY+YZ= XZ Addition property of equality
UW=XZ Substitution property of equality
The length of UW is congruent to the length of XZ Transitive Property of Segment Congruence
3)
Given: The length of AB is congruent to the length of JK, the length of JK is congruent to the length of ST
Prove: THe length of AB is congruent to the length of ST
Statement Reasoning
The length of AB is congruent to the length of JK, Given
The length of JK is congruent to the length of ST
AB = JK, JK =ST Definition of Congruent Segments
AB = ST Transitive property of equality
The length of AB is congruent to the length of ST Definition of congruent segments
4) Given: VN is 10, LA is 10
Prove : VN= LA
Statement Reasoning
VN is 10, LA is 10 Given
VN= LA Definition of equal segments
Reflexive Property
For any segment, AB=AB. Meaning that the measurement of a segment equals its measurement, obviously . Until now, the reflexive property has been the least used in our class probibly because it is the most obvious of the properties. Really, there is no need to prove that A=A, we know it already, it isn't complicated(doesn't need explaining).
SOME EXAMPLES:
MORE EXAMPLES...
1)
| EF = EF | Given | |||||||
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Definition of Congruent Segments |
2)
| CD = CD | Given | |||||||
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Definition of Congruent Segments |
3)
| DB = DB | Given | |||||||
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Definition of Congruent Segments |
4)
| ZT = ZT | Given | |||||||
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Definition of Congruent Segments |
Transitive Property
If a = b and b = c, then a = c
remember that if the points are in the same formula than this one the points all have the same degree. To prove the Transative property of Congruency for segment always begin by drawing the three congruent angles. And lebel them A,B,C or 1,2,3.
EXAMPLES...
GIVEN... A congruent C, B congruent C
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Statements |
Reasons |
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1. A congruent B B congruent to C |
1. Given |
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2. mA=mB |
2. Definition of congruent segment |
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3. mB=mC |
3. Definition of congruent segment |
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4. A=C |
4. Transitive Property
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5. A congruent C |
Definition of congruent segment |
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Statements |
Reasons |
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1. <B congruent to <A |
1. Given |
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2. <A congruent to <E |
2. Given |
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3. <B congruent to <E |
3. Transitive Property |
| TTT!!! |
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Statements |
Reasons |
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1. A congruent B B congruent to D |
1. Given |
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2. <A congruent to <D |
2. Transitive Property |
| 3. (<C +<A + <B + <D)/4= <C |
3. Division Property |
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TTT!!! |
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Statements |
Reasons |
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1. <A congruent to <C <A+<C=<B |
1. Given |
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2. <B=90 degrees |
2. Def of Right angle |
| 3. <D=180 degrees |
3. def of Straight angle |
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4. <A+<B+<C=<D |
4. Transitive
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5. <A+<A=<B |
Substitution |
| 6. 2<A=<B |
6. Multiplication |
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7. 2<A=90 degrees |
7. Substitution |
| 3. <A=45 degrees |
3. Division Property |
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TTT!!! |
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Statements |
Reasons |
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1. AB congruent JK, JK congruent ST |
1. Given |
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2. AB=JK, JK=ST |
2. Definition of congruent segment |
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3. AB=ST |
3. Transative property |
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4. AB congruent ST |
4. Definition of congruent segment |
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Statements |
Reasons |
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1.<B+<C=180 degrees |
1. Supplementary |
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2. <B congruent to <A |
2. Given |
| 3. <A congruent to <D |
3. Given |
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4. <B congruent to <D |
4. Transitive
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