Geometry Lessons 2.4, 2.5, and 2.6 Vocabulary and Proof Justifications

A set of 67 vocabulary flashcards covering algebraic properties of equality, segment and angle congruence properties, geometric definitions, postulates, theorems, and statement justifications for two-column proofs.

Two-Column Proof

A format to organize a proof with statements/steps on the left side (what) and reasons to justify each statement on the right side (why).

Addition Property of Equality

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

Subtraction Property of Equality

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

Multiplication Property of Equality

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

Division Property of Equality

If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

Substitution Property of Equality

If $a = b$, then $a$ can be replaced with $b$ in any equation or expression.

Distributive Property

For sum: $a(b + c) = ab + ac$; for difference: $a(b - c) = ab - ac$.

Reflexive Property of Equality

For any real number $a$, $a = a$.

Symmetric Property of Equality

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

Transitive Property of Equality

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

Reflexive Property of Segment Lengths

For any segment length $AB$, $AB = AB$.

Symmetric Property of Segment Lengths

If $AB = CD$, then $CD = AB$.

Transitive Property of Segment Lengths

If $AB = CD$ and $CD = EF$, then $AB = EF$.

Reflexive Property of Angle Measures

For any angle measure $m\angle A$, $m\angle A = m\angle A$.

Symmetric Property of Angle Measures

If $m\angle A = m\angle B$, then $m\angle B = m\angle A$.

Transitive Property of Angle Measures

If $m\angle A = m\angle B$ and $m\angle B = m\angle C$, then $m\angle A = m\angle C$.

Definition of Congruence

Segments or angles are congruent if and only if they have the same measure (e.g., $\overline{AB} \cong \overline{CD}$ if and only if $AB = CD$).

Definition of Midpoint

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

Definition of Angle Bisector

A ray, line, or segment that divides an angle into two congruent angles (if $\overrightarrow{YW}$ bisects $\angle XYZ$, then $\angle XYW \cong \angle ZYW$).

Reflexive Property of Segment Congruence

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

Symmetric Property of Segment Congruence

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

Transitive Property of Segment Congruence

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

Reflexive Property of Angle Congruence

For any angle $\angle A$, $\angle A \cong \angle A$.

Symmetric Property of Angle Congruence

If $\angle A \cong \angle B$, then $\angle B \cong \angle A$.

Transitive Property of Angle Congruence

If $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$.

Deductive Reasoning in Proofs

Using facts, definitions, accepted properties, and laws of logic to make logical statements one at a time until reaching the conclusion.

Reasons in Two-Column Proofs

Justifications used on the right side of a proof, which can only be properties, definitions, postulates, or theorems.

Segment Addition Postulate

If point $K$ is between points $J$ and $L$, then $JK + KL = JL$.

Definition of Right Angle

An angle is a right angle if and only if its measure is $90^\circ$.

Definition of Complementary Angles

Two angles are complementary if the sum of their measures is $90^\circ$ ($m\angle P + m\angle Q = 90^\circ$).

Definition of Supplementary Angles

Two angles are supplementary if the sum of their measures is $180^\circ$ ($m\angle W + m\angle X = 180^\circ$).

Definition of Perpendicular Lines

Two lines that intersect to form right angles (if $l \perp m$, then $\angle 1$ is a right angle).

Linear Pair Postulate

If two angles form a linear pair, then they are supplementary.

Angle Addition Postulate

If point $M$ lies in the interior of $\angle JKL$, then $m\angle JKM + m\angle MKL = m\angle JKL$.

Theorem

A statement that can be proven, and once proven, can be used as a reason in other proofs.

Right Angles Congruence Theorem

All right angles are congruent.

Congruent Supplements Theorem

If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.

Congruent Complements Theorem

If two angles are complementary to the same angle (or to congruent angles), then they are congruent.

Vertical Angle Congruence Theorem

Vertical angles are congruent.

Algebraic Proof Reason: $4x - 1 = 27 \rightarrow 4x = 28$

Addition Property of Equality.

Algebraic Proof Reason: $4x = 28 \rightarrow x = 7$

Division Property of Equality.

Algebraic Proof Reason: $-9(2x - 3) = 63 \rightarrow -18x + 27 = 63$

Distributive Property.

Algebraic Proof Reason: $-18x + 27 = 63 \rightarrow -18x = 36$

Subtraction Property of Equality.

Algebraic Proof Reason: $-18x = 36 \rightarrow x = -2$

Division Property of Equality.

Statement Justification: If $k = 3$, then $3 = k$

Symmetric Property of Equality.

Statement Justification: If $2x = 14$, then $x = 7$

Division Property of Equality.

Statement Justification: $10y = 10y$

Reflexive Property of Equality.

Statement Justification: If $-5x - 1 = -11$, then $-5x = -10$

Addition Property of Equality.

Statement Justification: If $10a = 2b$ and $2b = c$, then $10a = c$

Transitive Property of Equality.

Statement Justification: $-7(n - 4) = -7n + 28$

Distributive Property.

Statement Justification: If $6y = 24$, then $6y - 3 = 24 - 3$

Subtraction Property of Equality.

Statement Justification: If $10x + w = 41$ and $w = 1$, then $10x + 1 = 41$

Substitution Property of Equality.

Statement Justification: If $3x = 2y$ and $2y = z$, then $3x = z$

Transitive Property of Equality.

Statement Justification: If $7m = 35$, then $7m + 4 = 35 + 4$

Addition Property of Equality.

Proof Statement Justification: If $PQ = PQ$, then $\overline{PQ} \cong \overline{PQ}$

Definition of Congruence.

Proof Statement Justification: $\overline{EF} \cong \overline{EF}$

Reflexive Property of Segment Congruence.

Proof Statement Justification: If $AB = DE$, then $DE = AB$

Symmetric Property of Equality.

Proof Statement Justification: If $Y$ is the midpoint of $\overline{XZ}$, then $XY = YZ$

Definition of Midpoint.

Proof Statement Justification: If $\overline{FG} \cong \overline{HI}$ and $\overline{HI} \cong \overline{JK}$, then $\overline{FG} \cong \overline{JK}$

Transitive Property of Segment Congruence.

Proof Statement Justification: If $PQ + RS = TV$ and $RS = WX$, then $PQ + WX = TV$

Substitution Property of Equality.

Proof Statement Justification: If $LP = PN$, and $L, P, N$ are collinear, then $P$ is the midpoint of $\overline{LN}$

Definition of Midpoint.

Proof Statement Justification: If $\overline{UV} \cong \overline{UV}$, then $UV = UV$

Definition of Congruence.

Proof Statement Justification: If $RS = ST$ and $ST = 2UV$, then $RS = 2UV$

Transitive Property of Equality.

Proof Statement Justification: If $\angle C$ is a right angle, then $m\angle C = 90^\circ$

Definition of Right Angle.

Proof Statement Justification: If $m\angle P + m\angle Q = 90^\circ$, then $\angle P$ and $\angle Q$ are complementary

Definition of Complementary Angles.

Proof Statement Justification: If $l \perp m$, then $\angle 1$ is a right angle

Definition of Perpendicular Lines.

Proof Statement Justification: If $\angle W$ and $\angle X$ are supplementary, then $m\angle W + m\angle X = 180^\circ$

Definition of Supplementary Angles.