1-1 Properties of Real Numbers

Property

Just like numbers, congruent triangles follow fundamental properties of equality:

• Reflexive Property: $\Delta ABC \cong \Delta ABC$ (A triangle is congruent to itself)
• Symmetric Property: If $\Delta ABC \cong \Delta DEF$, then $\Delta DEF \cong \Delta ABC$
• Transitive Property: If $\Delta ABC \cong \Delta DEF$ and $\Delta DEF \cong \Delta PQR$, then $\Delta ABC \cong \Delta PQR$

Examples

• Reflexive: In a geometric proof where two triangles share a common wall (side $\overline{BD}$), you state $\overline{BD} \cong \overline{BD}$ by the Reflexive Property.
• Symmetric: If you prove $\Delta XYZ \cong \Delta LMN$, you can freely state $\Delta LMN \cong \Delta XYZ$ if it helps match the format of the question.
• Transitive: If Triangle 1 is a clone of Triangle 2, and Triangle 2 is a clone of Triangle 3, then Triangle 1 must be a clone of Triangle 3.

Explanation

These properties describe the fundamental rules of mathematical logic. The Reflexive Property is incredibly common in proofs when two triangles share a side or an angle. The Transitive Property acts as a logical bridge, allowing you to connect two separate figures by comparing them both to a common third figure.

Property

The identity property of addition: for any real number $a$,

$0$ is called the additive identity.

The identity property of multiplication: for any real number $a$,

$1$ is called the multiplicative identity.

Examples

• The expression $42 + 0$ simplifies to $42$ by the identity property of addition.
• Using the identity property of multiplication, the expression $1 \cdot (-5y)$ simplifies to $-5y$.
• When simplifying $0 + (x+y)$, the identity property of addition shows the result is just $x+y$.