Lesson 2-5
Other Algebraic Properties of Equality
AB  AB
Reflexive Property x  x
77
Objective – To provide reasons for statements
made in algebraic proof.
Proof - An argument based on logical deduction
that uses definitions, properties, and
theorems to show a conclusion is true.
Given 3(x  2)  21, show that x  5.
Statement
Reasons
1) 3(x  2)  21
2) 3x  6  21
3) 3x  15
4) x  5
Symmetric Property If x  y, then y  x.
If x  3, then 3  x.
If MN  AB, then AB  MN.
Transitive Property
If a  b and b  c, then a  c.
Given
Substitution Property
If a  b, then b can be substituted for a.
Distributive Property
Subtract Prop. of Equal.
If AB  XY and 10  AB  25, then
10  XY  25.
Division Prop. of Equal.
If AG  2x  8, GB  3x  7, and AB  6x  24,
A 2x  8 G 3x  7 B
find AG.
6x  24
Statement
1) AG  2x  8,GB  3x  7,
AB  6x  24
2) AG  GB  AB
3) 2x  8  3x  7  6x  24
4) 5x  1  6x  24
5) 1  x  24
6) 25  x
7) AG  2(25)  8
8) AG  58
Reasons
Statement
1) AC  BD
2) BC  BC
3) AC  BD
BC  BC
4) AB  BC  AC
BC  CD  BD
5) AB  BC  BC  CD
6) AB  CD
7) AB  CD
8) CD  AB
Symmetric Property of Congruence
Given
Segment Addition Post.
Substitution (1 into 2)
Simplify
Subtract Prop of Equal.
Add Prop of Equal.
Substitution (6 into 1)
Simplify
If AC  BD, prove CD  AB.
A
Properties of Congruence
Reflexive Property of Congruence
AB  AB
XY  YX
B
C
D
Reasons
Given
Reflexive Prop of Cong.
Def. of Congruence
Segment Addition Post.
Substitution (4 into 3)
Subtract Prop of Equal.
Def. of Congruence
Symmetric Prop of Cong.
If AB  XY, then XY  AB.
If ABC  LMN, then LMN  ABC.
Transitive Property of Congruence
If ABC PQR and PQR XYZ,
then ABC XYZ.
If AEB  CED, prove BED  AEC.
Statement
Reasons E
1) AEB  CED
2) BEC  BEC
3) mAEB  mCED
mBEC  mBEC
A
B
C
Given
D
Reflexive Prop of Cong.
Def. of Congruence
4) mAEB  mBEC 
Add Prop of Equality
mCED  mBEC
5) mAEB  mBEC  mAEC Angle Addition Post.
mCED  mBEC  mBED
6) mAEC  mBED
7) AEC  BED
8) BED  AEC
Substitution (5 into 4)
Def. of Congruence
Symmetric Prop of Cong.
Geometry Slide Show: Teaching Made Easy As Pi, by James Wenk © 2014
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