REASONING IN ALGEBRA PROPERTIES OF EQUALITY AND CONGRUENCE
Reflexive Property
The number equal to itself is called reflexive property in algebra.
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Equality AB = AB m∠A = m∠A |
Congruence AB ≅ AB ∠A ≅ ∠A |
Symmetric Property
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Equality If AB = CD, then CD = AB. If m∠A = m∠B, then m∠B = m∠A. |
Congruence If AB ≅ CD, then CD ≅ AB. If ∠A ≅ ∠B, then ∠B ≅ ∠A. |
Transitive Property
Equality
If AB = CD and CD = EF, then AB = EF.
If m∠A = m∠B and m∠B = m∠C, then m∠A = m∠C.
Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF.
If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.
PROPERTIES OF EQUALITY
Addition Property
Adding the same number to each side of an equation produces an equivalent equation.
Example
x - 3 = 7
x - 3 + 3 = 7 + 3
Subtraction Property
Subtracting the same number from each side of an equation produces an equivalent equation.
Example
y + 5 = 11
y + 5 - 5 = 11 - 5
Multiplication Property
Multiplying each side of an equation by the same nonzero number produces an equivalent equation.
Example
Problem 1 :
Write the letter of each property next to its definition. The letters a, b, and c represent real numbers.
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1. If a = b, then b = a 2. If a = b, then ac = bc 3. AB ≅ AB 4. a = a 5. If a = b, then a + c = b + c 6. a(b + c) = ab + ac 7. If a = b and b = c, then a = c 8. If ∠P ≅ ∠Q, then ∠Q ≅ ∠P 9. If ∠A ≅ ∠B, and ∠B ≅ ∠C, then ∠A ≅ ∠C 10. If a = b and c ≠ 0, then a/c = b/c 11. If a = b, then b can be substituted for a in any expression 12. If a = b, then a - c = b - c |
A. Addition Property of Equality B. Subtraction Property of Equality C. Multiplication Property of Equality D. Division Property of Equality E. Reflexive Property of Equality F. Symmetric Property of Equality G. Transitive Property of Equality H. Substitution Property of Equality I. Distributive Property J. Reflexive Property of congruence K. Symmetric Property of Congruence L. Transitive Property of Congruence |
Solution:
1. If a = b, then b = a. This is symmetric Property of Equality
2. If a = b, then ac = bc. This is Multiplication Property of Equality
3. AB ≅ AB. This is Reflexive Property of Congruence
4. a = a. This is Reflexive Property of Equality
5. If a = b, then a + c = b + c. This is Addition Property of Equality
6. a(b + c) = ab + ac. This is Distributive Property
7. If a = b and b = c, then a = c. This is Transitive Property of Equality
8. If ∠P ≅ ∠Q, then ∠Q ≅ ∠P. This is Symmetric Property of Congruence
9. If ∠A ≅ ∠B, and ∠B ≅ ∠C, then ∠A ≅ ∠C. This is Transitive Property of Congruence
10. If a = b and c ≠ 0, then a/c = b/c. This is Division Property of Equality
11. If a = b, then b can be substituted for a in any expression. This is Substitution Property of Equality
12. If a = b, then a - c = b - c. This is Subtraction Property of Equality
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1. If a = b, then b = a 2. If a = b, then ac = bc 3. AB ≅ AB 4. a = a 5. If a = b, then a + c = b + c 6. a(b + c) = ab + ac 7. If a = b and b = c, then a = c 8. If ∠P ≅ ∠Q, then ∠Q ≅ ∠P 9. If ∠A ≅ ∠B, and ∠B ≅ ∠C, then ∠A ≅ ∠C 10. If a = b and c ≠ 0, then a/c = b/c 11. If a = b, then b can be substituted for a in any expression 12. If a = b, then a - c = b - c |
F. Symmetric Property of Equality C. Multiplication Property of Equality J. Reflexive Property of congruence E. Reflexive Property of Equality A. Addition Property of Equality I. Distributive Property G. Transitive Property of Equality K. Symmetric Property of Congruence L. Transitive Property of Congruence D. Division Property of Equality H. Substitution Property of Equality B. Subtraction Property of Equality |
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