Geometry Unit 2: Parallel Lines, Definitions, Postulates, and Proofs

Comprehensive practice flashcards covering Geometry Unit 2 vocabulary, definitions, postulates, equality/congruence properties, parallel line theorems, and proofs reasons.

Under what condition may the Properties of Equality be applied in geometric proofs compared to the Properties of Congruence?

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

What do the acronyms APOE, SPOE, MPOE, and DPOE stand for?

APOE stands for Addition Property of Equality, SPOE stands for Subtraction Property of Equality, MPOE stands for Multiplication Property of Equality, and DPOE stands for Division Property of Equality.

What is the Definition of Congruent Segments?

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

What is the Definition of Congruent Angles?

$m\angle X = m\angle Y$ if and only if $\angle X \cong \angle Y$.

What does the Reflexive Property of Congruence state for segments and angles?

For any segment, $\overline{AB} \cong \overline{AB}$; for any angle, $\angle X \cong \angle X$.

What does the Symmetric Property of Congruence state for segments and angles?

For segments, if $\overline{AB} \cong \overline{CD}$, then $\overline{CD} \cong \overline{AB}$; for angles, if $\angle X \cong \angle Y$, then $\angle Y \cong \angle X$.

What does the Transitive Property of Congruence state for segments and angles?

For segments, if $\overline{AB} \cong \overline{CD}$ and $\overline{CD} \cong \overline{EF}$, then $\overline{AB} \cong \overline{EF}$; for angles, if $\angle X \cong \angle Y$ and $\angle Y \cong \angle Z$, then $\angle X \cong \angle Z$.

What is the Definition of Midpoint?

The midpoint of a segment divides the segment into $2$ equal (congruent) parts (e.g., if $M$ is the midpoint of $\overline{AB}$, then $AM = MB$).

What is the Definition of a Segment Bisector?

A bisector intersects a segment at its midpoint.

What does the Segment Addition Postulate state?

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

What is the Definition of an Angle Bisector?

An angle bisector divides an angle into two equal (congruent) parts.

What is the Definition of Complementary Angles?

Two angles are complementary if and only if the sum of their measures is $90^\circ$.

What is the Definition of Supplementary Angles?

Two angles are supplementary if and only if the sum of their measures is $180^\circ$.

What is the Definition of Perpendicular?

Perpendicular lines form right angles (if $\overline{AB} \perp \overline{CD}$, then they intersect to form a right angle).

What is the Definition of a Right Angle?

If $\angle A$ is a right angle, then $m\angle A = 90^\circ$.

What is the Definition of a Linear Pair?

If two angles form a linear pair, then they are adjacent angles whose non-common sides are opposite rays (forming a straight angle).

What does the Linear Pair Postulate (Supplement Theorem) state?

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

What does the Angle Addition Postulate state?

If $R$ is in the interior of $\angle PQS$, then $m\angle PQR + m\angle RQS = m\angle PQS$ (or $m\angle ABD + m\angle DBC = m\angle ABC$).

What does the Vertical Angles Theorem state?

If two angles are vertical, then they are congruent (if $\angle A$ and $\angle B$ are vertical, then $\angle A \cong \angle B$).

What does the Complement Theorem state?

If two angles form a right angle, then they are complementary.

What does the Congruent Complements Theorem state?

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

What does the Congruent Supplements Theorem state?

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

What does the Corresponding Angles Postulate state?

If two parallel lines are cut by a transversal, then each pair of corresponding angles is congruent.

What does the Converse of the Corresponding Angles Postulate state?

If two lines are cut by a transversal so that each pair of corresponding angles is congruent, then the lines are parallel.

What does the Alternate Interior Angles Theorem state?

If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent.

What does the Converse of the Alternate Interior Angles Theorem state?

If two lines are cut by a transversal so that each pair of alternate interior angles is congruent, then the lines are parallel.

What does the Alternate Exterior Angles Theorem state?

If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent.

What does the Converse of the Alternate Exterior Angles Theorem state?

If two lines are cut by a transversal so that each pair of alternate exterior angles is congruent, then the lines are parallel.

What does the Consecutive (Same-Side) Interior Angles Theorem state?

If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary.

What does the Converse of the Consecutive (Same-Side) Interior Angles Theorem state?

If two lines are cut by a transversal so that each pair of consecutive interior angles is supplementary, then the lines are parallel.

What does the Consecutive (Same-Side) Exterior Angles Theorem state?

If two parallel lines are cut by a transversal, then each pair of consecutive exterior angles is supplementary.

What does the Converse of the Consecutive (Same-Side) Exterior Angles Theorem state?

If two lines are cut by a transversal so that each pair of consecutive exterior angles is supplementary, then the lines are parallel.

What does the Perpendicular Transversal Theorem state?

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

What does the Converse of the Perpendicular Transversal Theorem state?

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