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PROPERTIES OF EQUALITY
In Geometry deductive reasoning is used to prove conjectures and theorems. In this unit you will begin to examine a logical procedure for verifying geometric relationships. You will start with reviewing the “properties of equality” for real numbers, and then relate them to geometric measures. You will apply the reflexive, symmetric, transitive, and substitution properties to provide reasons for geometric statements and to write informal proofs.
Real Number Properties of Equality – Chart
Reflexive and Symmetric Properties
Transitive and Substitution Properties
Informal Proof
Real Number Properties of Equality
In the table below, each of the properties of equality for real numbers are listed with a general explanation of each property. The properties are used to verify steps in a proof.
Properties of Equality for Real Numbers
Reflexive Property For every number a , a = a.
Symmetric Property For all numbers a and b , if a = b , then b = a. Transitive Property For all numbers a , b , and c , if a = b and b = c , then a = c. Addition and Subtraction Properties
For all numbers a , b , and c , if a = b , then a + c = b + c and a – c = b – c.
Multiplication and Division Properties
For all numbers a , b , and c , if a = b , then a c ⋅ = b c ⋅ ,
and if c ≠ 0,
a b c c
Substitution Property For all numbers a and b , if a = b , then a may be replaced by b in any equation or expression.
Distributive Property For all numbers a , b , and c , a ( b + c ) = ab + ac.
Measures of segments and angles are real numbers; thus, the properties of equality may be used to show many relationships in geometry.
Reflexive and Symmetric Properties
Reflexive Property for Segments
In the figure, QS is shared by + QRS and + QTS. In proofs which involve geometric figures such as this one, the reflexive property is used to illustrate that a segment is congruent to itself.
The red hash marks mean congruency.
QR is congruent to QT , denoted by one hash mark on each segment.
RS is congruent to TS , denoted by two hash marks on each congruent segment.
QS is congruent to QS , itself, denoted by three hash marks on that segment.
*Note: QS is shared by + QRS and + QTS
Q
R
S
T
Postulate 4-A Reflexive Property
Any segment or angle is congruent to itself. QS ≅ QS
Reflexive Property for Angles
In the figure, ∠ ABD is shared by + ABC and + DBE. In proofs which involve geometric figures such as this one, we will use the reflexive property to illustrate that an angle is congruent to itself.
The curved red hash marks mean congruency. The corner notch (appears to be square in shape) is a mark for denoting right angles.
∠ ACB and ∠ DEB are right angles and each are marked with the corner notch for right angles. Thus, they are congruent.
∠ A ≅ ∠ D. These congruent angles are denoted by one curved hash mark on each angle.
∠ B is congruent to ∠ B , itself, ( ∠ B ≅ ∠ B )and marked with two curved hash marks. Note: angle B is shared by both +^ ABC and + DBE.
Property of Symmetry
Postulate 4-B Symmetric Property
If AB ≅ CD , then CD ≅ AB. If ∠ CAB ≅ ∠ DOE , then ∠ DOE ≅ ∠ CAB.
A
B
C
D
E
Example : Given: ∠ A ≅ ∠ B. Find the measurement of each angle.
Congruent angles have equal measures by definition of congruency. Therefore, the expression for m ∠ A can be set equal to the expression for the m ∠ B.
8 13 3 2 13 8 3 15
x x
x x
x x
x x
By the substitution property, 3 can be substituted in for x to find the measure of each angle. It is a good idea to substitute the value for x into both expressions even though the measures of the angles are given as equal. Use the expression for ∠ B as a check to verify that both measurements are the same.
The measure of angle B verifies the measure of angle A.
Each angle measures 11 .°
A
8 x - 13
B
3 x^ + 2
A x B x A B A B
Informal Proof
Informal Proof (paragraph proof) – An informal proof is a paragraph of statements that explain why a conjecture is true.
Example 1 : Write a paragraph proof to support the conclusion based on the given information.
Given: 1and are supplementary; 2 Conclusion: and are supplementary.
Using the substitution property, ∠ 3 may be substituted for ∠ 2 since congruent angles have the same measure. Therefore, ∠1 and ∠ 3 are supplementaryas illustrated below.
