Exploring the Fundamentals: Properties of Equality and Congruence in Geometry

• Reflexive Property: For any quantity \( a\), \( a = a \).
• Symmetric Property: If \( a = b \), then \( b = a \).
• Transitive Property: If \( a = b\) and \( b = c \), then \( a = c \).
• Addition Property: If \( a = b \), then \( a + c = b + c \).
• Subtraction Property: If \( a = b \), then \( a – c = b – c \).
• Multiplication Property: If \( a = b \), then \( ac = bc \).
• Division Property: If \( a = b \) and \( c ≠ 0 \), then \( \frac{a}{c} = \frac{b}{c} \).

• Reflexive Property: Any geometric figure is congruent to itself. For any segment \( AB \), \( AB \cong AB \).
• Symmetric Property: If segment \( AB \cong CD \), then segment \ CD \cong AB \).
• Transitive Property: If \( AB \cong CD \) and \( CD \cong EF \), then \( AB \cong EF \).

Examples

Practice Questions:

1. If \( a = b \) and \( b = 7 \), what is \( a \) based on the properties of equality?
2. Given segment \( XY \cong ST \) and segment \( ST \cong UV \), what can you conclude about segments \( XY \) and \( UV \)?
3. If two angles are each congruent to \( 45^\circ \), are the two angles congruent to each other?

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1. \( a = 7 \) (By the Transitive Property of Equality)
2. Segment \( XY \) is congruent to segment \( UV \) (By the Transitive Property of Congruence).
3. Yes, the two angles are congruent to each other (By the Transitive Property of Equality).

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