Geometry Proofs, Properties of Equality, Properties of Congruence, and Logic Statements

Comprehensive practice flashcards covering segment proofs, properties of equality, properties of congruence, definitions, postulates, and conditional logic statements with converse, inverse, and contrapositive analysis.

What is the algebraic definition of the Addition Property of Equality?

If $a = b$, then $a + c = b + c$.

What is the algebraic definition of the Subtraction Property of Equality?

If $a = b$, then $a - c = b - c$.

What is the algebraic definition of the Multiplication Property of Equality?

If $a = b$, then $a \cdot c = b \cdot c$.

What is the algebraic definition of the Division Property of Equality?

If $a = b$, then $a \div c = b \div c$.

What is the Distributive Property?

For $a(b + c)$, $a(b + c) = ab + ac$.

What is the Substitution Property of Equality?

If $a = b$, then $a$ may be replaced by $b$ in any expression or equation.

What is the Reflexive Property of Equality?

For any real number $a$, $a = a$ (a value always equals itself).

What is the Symmetric Property of Equality?

If $a = b$, then $b = a$.

What is the Transitive Property of Equality?

If $a = b$ and $b = c$, then $a = c$.

Which property justifies the statement: 'If $k = 3$, then $3 = k$'?

Symmetric Property

Which property justifies the statement: 'If $2x = 14$, then $x = 7$'?

Division Property

Which property justifies the statement: '$10y = 10y$'?

Reflexive Property

Which property justifies the statement: 'If $-5x - 1 = -11$, then $-5x = -10$'?

Addition Property

Which property justifies the statement: 'If $10a = 2b$ and $2b = c$, then $10a = c$'?

Transitive Property

Which property justifies the statement: '$-7(n - 4) = -7n + 28$'?

Distributive Property

Which property justifies the statement: 'If $6y = 24$, then $6y - 3 = 24 - 3$'?

Subtraction Property

Which property justifies the statement: 'If $10x + w = 41$ and $w = 1$, then $10x + 1 = 41$'?

Substitution Property

Which property justifies the statement: 'If $3x = 2y$ and $2y = z$, then $3x = z$'?

Transitive Property

Which property justifies the statement: 'If $7m = 35$, then $7m + 4 = 35 + 4$'?

Addition Property

Which property justifies the statement: 'If $-2c = 18$, then $18 = -2c$'?

Symmetric Property

Which property justifies the statement: 'Given $3x^2 + 1$, if $x = 5$, then $3(5)^2 + 1$'?

Substitution Property

Which property justifies the statement: 'If $m = -2$, then $8m = -16$'?

Multiplication Property

Which property justifies the statement: '$5x + 8x = x(5 + 8)$'?

Distributive Property

When are Properties of Equality used versus Properties of Congruence?

Properties of Equality may only be used with equal signs ($=$), whereas Properties of Congruence are applied to statements with congruence symbols ($\cong$).

What is the Reflexive Property of Congruence?

For any segment $\overline{AB}$, $\overline{AB} \cong \overline{AB}$.

What is the Symmetric Property of Congruence?

If $\overline{AB} \cong \overline{CD}$, then $\overline{CD} \cong \overline{AB}$.

What is the Transitive Property of Congruence?

If $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$, then $\overline{AB} \cong \overline{EF}$.

What is the Definition of Congruence for line segments?

Segments are congruent if and only if they have the same measure: If $AB = CD$, then $\overline{AB} \cong \overline{CD}$, and if $\overline{AB} \cong \overline{CD}$, then $AB = CD$.

What is the Definition of Midpoint?

The midpoint of a segment divides the segment into two congruent parts (equal lengths). If $M$ is the midpoint of $\overline{AB}$, then $\overline{AM} \cong \overline{MB}$.

What is the Segment Addition Postulate?

If $A$, $B$, and $C$ are collinear points and $B$ is between $A$ and $C$, then $AB + BC = AC$.

In segment proof 27, given $Q$ is the midpoint of $\overline{PR}$ and $R$ is the midpoint of $\overline{QS}$, what reason justifies statement 3 ($PQ = QR$; $QR = RS$)?

Definition of congruence

In segment proof 27, what reason justifies statement 5 ($PQ + PQ = PQ + RS$)?

Segment Addition Postulate

In segment proof 27, what reason justifies statement 8 ($QR + RS = QS$)?

Segment Addition Postulate

In segment proof 28, given $\overline{AB} \cong \overline{CD}$ and $\overline{CE} \cong \overline{AE}$, what reason justifies statement 2 ($AB = CD$; $CE = AE$)?

Definition of congruence

In segment proof 28, what reason justifies statement 3 ($AE + EB = AB$; $CE + ED = CD$)?

Segment Addition Postulate

In segment proof 28, what reason justifies statement 4 ($CE + EB = CD$)?

Substitution Property

In segment proof 28, what reason justifies statement 6 ($ED = EB$)?

Subtraction Property

In segment proof 28, what reason justifies statement 7 ($\overline{ED} \cong \overline{EB}$)?

Definition of congruence

What are the converse, inverse, and contrapositive of the conditional statement 'If you live in New Jersey, then you live on the east coast', and what are their truth values?

Converse: 'If you live on the east coast, then you live in NJ' (False). Inverse: 'If you do not live in NJ, then you do not live on the east coast' (False). Contrapositive: 'If you do not live on the east coast, then you do not live in NJ' (True).

What are the converse, inverse, and contrapositive of 'If a polygon is a rectangle, then it is a square', and what are their truth values?

Conditional: False. Converse: 'If a polygon is a square, then it is a rectangle' (True). Inverse: 'If a polygon is not a rectangle, then it is not a square' (True). Contrapositive: 'If a polygon is not a square, then it is not a rectangle' (False).

What are the truth values for the conditional statement 'If 2 angles have a sum of $180$, then they are supplementary' and its converse, inverse, and contrapositive?

All four statements (Conditional, Converse, Inverse, and Contrapositive) are True.

What are the converse, inverse, and contrapositive of 'If 2 angles have a sum of $90$, then both angles are acute', and what are their truth values?

Conditional: True. Converse: 'If both angles are acute, then they have a sum of $90$' (False). Inverse: 'If 2 angles do not have a sum of $90$, then both angles are not acute' (False). Contrapositive: 'If both angles are not acute, then they do not have a sum of $90$' (True).