Comprehensive Study Guide: Foundations of Geometry, Relations, Functions, and Quadrilateral Properties

Foundations of Geometry: Points, Lines, Rays, Segments, and Planes

• Point

  • Representation: Drawn as a single dot.

  • Naming Convention: Named using a single capital letter (e.g., Point $P$).

  • Representation / Symbol: Point $P$

• Line

  • Representation: Drawn as a straight stroke with an arrowhead at each end to signify infinite extension in two opposite directions.

  • Naming Convention: Named using a lowercase script letter (e.g., line $m$) or by any two points lying on the line with a double-sided arrow over the letters (e.g., line $AB$, line $BA$, $\overleftrightarrow{AB}$, or $\overleftrightarrow{BA}$ ).

• Plane

  • Definition: A plane is a flat surface that extends without end in all directions.

  • Representation: Drawn as a shaded four-sided figure.

  • Naming Convention: Named using a capital script letter (e.g., plane $T$) or by naming three non-collinear points lying on the plane in any order (e.g., plane $XYZ$, plane $XZY$, plane $YXZ$, plane $YZX$, plane $ZXY$, plane $ZYX$).

• Line Segments

  • Definition: A line segment is a part of a line that has a definite start and end. A segment contains exactly two endpoints.

  • Representation: Drawn with two points (endpoints) marking both ends of the segment.

  • Naming Convention: Named using its two endpoints in either order.

  • Symbols / Words: line segment $AB$, line segment $BA$, $\overline{AB}$, or $\overline{BA}$.

  • Equality Note: $\overline{AB}$ and $\overline{BA}$ represent the exact same line segment.

• Rays

  • Definition: A ray is a part of a line that has a definite starting point (called the endpoint) and extends without end in one direction.

  • Representation: Drawn with one point representing the starting endpoint and an arrowhead at the opposite end.

  • Naming Convention: Named starting with the endpoint first, followed by any other point on the ray in the direction of the arrow.

  • Symbols / Words: ray $AB$, ray $BA$, $\overrightarrow{AB}$ , or $\overrightarrow{BA}$.

  • Directionality Note: $\overrightarrow{AB}$ and $\overrightarrow{BA}$ are two completely different rays. The first letter in the symbol denotes the initial endpoint of the ray.

  • Structural Note: Rays and line segments are geometric subsets/parts of lines.

Angle Relationships and Definitions

• Opposite Rays

  • Definition: Two rays that lie on the exact same line, extend in opposite directions, and share only a single common endpoint.

  • Example: Rays $\overrightarrow{CA}$ and $\overrightarrow{CB}$ are opposite rays because both reside on the same line and share the common endpoint $C$.

• Angles

  • Definition: When two rays share a common endpoint but are not part of the same line, the figure formed is called an angle.

  • Components of an Angle:

  • Vertex: The common endpoint shared by the two rays forming the angle.

  • Sides: The two non-collinear rays that make up the angle.

  • Detailed Example: For an angle formed by vertex $U$ and rays extending to points $F$ and $E$, $U$ is the vertex, while rays $\overrightarrow{UF}$ and $\overrightarrow{UE}$ form the sides of the angle.

  • Naming Conventions for Angles:

  • By three capital letters, with the vertex letter placed strictly in the middle (e.g., $\angle NUF$ or $\angle FUN$).

  • By the vertex letter only, provided there is only one angle located at that vertex (e.g., $\angle U$).

  • By a designated number assigned inside the interior of the angle (e.g., $\angle 1$).

• Angle Pairs

  • Adjacent Angles: Two angles that lie in the same plane, share a common vertex and a common side, but have no internal points in common.

  • Complementary Angles: Two angles whose sum of measures equals $90^\circ$.

  • Supplementary Angles: Two angles are supplementary if and only if the sum of their measures equals $180^\circ$.

  • Linear Pair: Two angles form a linear pair if and only if they are adjacent and their non-common sides form opposite rays.

  • Vertical Angles: Two angles are vertical angles if and only if their sides form two pairs of opposite rays. Vertical angles are always congruent.

  • Intersecting Lines: Lines that connect or cross each other at a single point.

  • Perpendicular Lines: Lines that cross or intersect each other to form right angles ($90^\circ$).

  • Parallel Lines: Coplanar lines that do not intersect at any point when extended infinitely.

Geometric Construction Tools and Methods

• Geometric Construction

  • Definition: The process of drawing accurate geometric figures using a strict, limited set of tools, traditionally a straightedge and a compass.

• Tools for Construction

  • Straightedge: An unlined tool used specifically to draw straight lines. Real-world objects that can serve as a straightedge include a ruler, a credit card or ATM card, or a sturdy piece of cardboard.

  • Compass: A specialized tool used for drawing circles, circular arcs, and marking off equal linear distances in geometric constructions.

• Geometric Construction vs. Drawing

  • Geometric Construction: Relies on specific tools and techniques; emphasizes exact mathematical accuracy and precision; used directly in geometric proof and problem solving.

  • Drawing: Utilizes freehand sketching; aims for a general representation of a figure; primarily used for visual illustration and conceptual visualization.

Lines, Transversals, and Angle Pairs

• Transversal

  • Definition: A line that intersects two or more coplanar lines at distinct, different points.

• Angle Pair Classifications Formed by a Transversal

  • Corresponding Angles: Angles that lie on the same side of the transversal and in corresponding positions relative to the two lines.

  • Alternate Interior Angles: Non-adjacent interior angles that lie on opposite sides of the transversal between the two cut lines.

  • Alternate Exterior Angles: Angles that lie on opposite sides of the transversal and are positioned outside the two cut lines.

  • Same-Side Interior Angles: Angles that lie on the same side of the transversal and are positioned inside between the two cut lines.

• Parallel Lines Cut by a Transversal Theorems

  • If two parallel lines are cut by a transversal, then:     a. Corresponding angles are congruent.     b. Alternate interior angles are congruent.     c. Alternate exterior angles are congruent.     d. Each pair of same-side interior angles is supplementary (sum equals $180^\circ$).     e. Each pair of same-side exterior angles is supplementary (sum equals $180^\circ$).

  • Example Expression: $(2x + 20)^\circ$

Relations, Functions, and the Coordinate System

• Relation

  • Definition: A set of ordered pairs. An ordered pair, commonly known as a point, consists of two components: the $x$-coordinate (input) and the $y$-coordinate (output), written as $(x, y)$ (e.g., $(2, 1)$).

• Function

  • Definition: A well-defined relation where no two distinct ordered pairs have the exact same first element ($x$-value). A function pairs every input in the domain with exactly one output in the range.

  • Essential Rule: There are no duplicate first elements ($x$-values) paired with differing second elements.

  • Four Ways to Represent Relations and Functions:

  1. Set of ordered pairs

  2. Correspondence or mapping diagram

  3. A graph on a coordinate plane

  4. An algebraic equation or rule

• Domain

  • Definition: The set of all first elements ($x$-values) in a relation. The domain represents the independent variable.

• Mapping Examples

  • Mapping Set: $\{(4, \text{PS}), (8, \text{PC}), (27, \text{PC}), (36, \text{PS}), (64, \text{PC})\}$

  • Input $4 \rightarrow \text{PERFECT SQUARE (PS)}$

  • Input $8 \rightarrow \text{PERFECT CUBE (PC)}$

  • Input $27 \rightarrow \text{PERFECT CUBE (PC)}$

  • Input $36 \rightarrow \text{PERFECT SQUARE (PS)}$

  • Input $64 \rightarrow \text{PERFECT CUBE (PC)}$

  • Evaluation: Because every element in the domain is matched to exactly one output, this relation defines a function.

• Vertical Line Test

  • Definition: A visual geometric method used to determine whether a graph on a coordinate plane represents a function.

  • Geometrical Rule: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

  • Step-by-Step Procedure:

  1. Imagine a vertical line passing through the graph.

  2. Count the number of points where the vertical line intersects the graph.

  3. Analyze the result:

     • If the vertical line intersects the graph at most once everywhere: It IS a function.

     • If the vertical line intersects the graph at two or more points anywhere: It is NOT a function.

• Algebraic Problem (Function Definition)

  • Problem: For what value of $k$ will the relation $R = \{(5k + 3, 4), (2k + 12, 9)\}$ NOT be a function?

  • Solution Steps:

  • A relation fails to be a function if the same input ($x$-value) yields two different outputs ($4 \neq 9$).

  • Set the input expressions equal to each other:       $5k + 3 = 2k + 12$       $5k - 2k = 12 - 3$       $3k = 9$       $k = 3$

  • Result: When $k = 3$, the relation contains $(18, 4)$ and $(18, 9)$, so it is NOT a function.

• Variables

  • Definition: A letter or symbol representing an unknown number, value, or quantity.

  • Dependent Variable: A variable whose value depends on and changes in response to another variable (typically $y$).

  • Independent Variable: A variable whose value can be chosen freely and does not depend on another variable (typically $x$ or $n$).

  • Example Equation: $y = 2x + 3$ ($y$ is the dependent variable, $x$ is the independent variable).

  • Example Context: Let $n$ be the number of chocolate bars purchased, where each chocolate bar ($c$) costs $24.00\,\text{pesos}$. The total cost depends on $n$.

Linear Functions, Slope, and Line Equations

• The Coordinate Plane

  • Terminology: Referred to as the Rectangular Coordinate System, $R^2$, the $xy$-plane, or the Cartesian plane.

  • Historical Origin: Named after René Descartes (1596–1650), the French mathematician known as the Father of Modern Mathematics.

  • Mathematical Definition: A two-dimensional flat surface defined by two perpendicular axes that extends infinitely in every direction.

• Linear Function

  • Definition: A function whose graph produces a non-vertical straight line.

• Slope of a Line

  • Definition: The measure of the steepness and direction of a line, represented by the variable $m$. It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points $(x_1, y_1)$ and $(x_2, y_2)$.

  • Slope Formula:     $m = \frac{\text{rise}}{\text{run}} = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$

  • Four Types of Slope:

  1. Positive Slope (m > 0): The line climbs upward from left to right.

  2. Negative Slope (m < 0): The line falls downward from left to right.

  3. Zero Slope ($m = 0$): A perfectly horizontal line ($y_2 - y_1 = 0$).

  4. Undefined Slope: A vertical line, because the horizontal change is zero ($x_2 - x_1 = 0$), leading to division by zero.

  • Intercepts: The points where a line crosses the coordinate axes (the $x$-intercept and $y$-intercept).

Conditions for Parallelism, Perpendicularity, and Coplanarity

• Parallel Lines Conditions

  • Definition: Two lines are parallel if and only if they are coplanar and do not intersect.

  • Parallel Line Postulate: Given a line and a point not on the given line, there exists exactly one line through the given point parallel to the given line.

  • Three Parallel Lines Theorem: If two lines are parallel to a third line, then they are parallel to each other.

  • Slope Rule: Two non-vertical lines are parallel if and only if they have equal slopes ($m_1 = m_2$).

• Perpendicular Lines Conditions

  • Definition: Two lines that intersect to form four right angles ($90^\circ$).

  • Perpendicular Bisector: A line, segment, or ray perpendicular to a segment at its precise midpoint.

  • Theorem: If two lines are perpendicular to each other, then they intersect to form right angles.

  • Slope Rule: Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals of each other ($m_1 \cdot m_2 = -1$).

• Worked Slope Example

  • Question: Are the lines $L_1$ (passing through $(-2, 1)$ and $(4, 5)$) and $L_2$ (passing through $(3, 0)$ and $(0, -2)$) parallel?

  • Calculation for $L_1$:     $m_1 = \frac{5 - 1}{4 - (-2)} = \frac{4}{6} = \frac{2}{3}$

  • Calculation for $L_2$:     $m_2 = \frac{-2 - 0}{0 - 3} = \frac{-2}{-3} = \frac{2}{3}$

  • Conclusion: Since $m_1 = m_2 = \frac{2}{3}$, the lines $L_1$ and $L_2$ are parallel ($L_1 \parallel L_2$).

• Coplanar and Non-Coplanar Lines

  • Coplanar Lines: Lines that lie on the exact same plane.

  • Non-Coplanar Lines: Lines that do not lie on the same plane.

  • Fundamental Geometric Facts:

  • Any two points are collinear and coplanar.

  • Any three points are always coplanar.

  • Two parallel lines are always coplanar.

Introduction to Quadrilaterals and Polygons

• Polygon

  • Definition: A closed plane figure formed by three or more line segments intersecting only at their endpoints.

  • Sides: The individual line segments forming the polygon.

  • Vertices: Points of intersection between any two consecutive sides.

  • Diagonals: Line segments connecting non-consecutive (opposite) vertices of a polygon.

• Quadrilateral

  • Definition: A four-sided polygon containing 4 sides, 4 interior angles, and 4 vertices.

  • Classification: Can be regular (all sides and angles equal) or irregular.

  • Angle Sum Theorem: The sum of all interior angles in any quadrilateral is strictly $360^\circ$.

  • Symbol: The symbol $\square$ denotes a quadrilateral.

  • Naming Rule: Named by listing consecutive vertices in order around the perimeter (e.g., $\square TEAL$).

  • Equivalent Naming Sequences for a Single Quadrilateral (e.g., $\square ABCD$):     $\square DCBA, \square ABCD, \square BCDA, \square CBAD, \square CDAB, \square BADC, \square DABC, \square ADCB$

• Quadrilateral Family Tree & Hierarchy

  • Polygons $\rightarrow$ Quadrilaterals

  • No Parallel Sides: Kite (contains 2 pairs of adjacent congruent sides).

  • 1 Pair of Parallel Sides: Trapezoid $\rightarrow$ Isosceles Trapezoid.

  • 2 Pairs of Parallel Sides: Parallelograms:

    • Rectangle (4 right angles)

    • Rhombus (4 congruent sides)

    • Square (4 congruent sides and 4 right angles)

Parallelograms: Definition, Properties, and Conditions

• Parallelogram

  • Definition: A quadrilateral with two pairs of parallel sides.

  • Parts of Parallelogram $\square ABCD$:

  • Opposite Sides: $\overline{AB}$ and $\overline{CD}$; $\overline{BC}$ and $\overline{AD}$.

  • Opposite Angles: $\angle A$ and $\angle C$; $\angle B$ and $\angle D$.

  • Consecutive Angles: $\angle A$ and $\angle B$; $\angle B$ and $\angle C$; $\angle C$ and $\angle D$; $\angle D$ and $\angle A$.

  • Diagonals: $\overline{AC}$ and $\overline{BD}$.

  • Base ($b$) and Altitude/Height ($h$): Height is the perpendicular distance to the base.

• Six Fundamental Properties of All Parallelograms

  1. Opposite sides are parallel ($\overline{AD} \parallel \overline{BC}$ and $\overline{AB} \parallel \overline{DC}$).

  2. Opposite sides are congruent ($\overline{AD} \cong \overline{BC}$ and $\overline{AB} \cong \overline{DC}$).

  3. Opposite angles are congruent ($\angle A \cong \angle C$ and $\angle B \cong \angle D$).

  4. Consecutive angles are supplementary:      $m\angle A + m\angle B = 180^\circ$      $m\angle B + m\angle C = 180^\circ$      $m\angle C + m\angle D = 180^\circ$      $m\angle D + m\angle A = 180^\circ$

  5. Diagonals bisect each other (if diagonals intersect at $E$, then $AE = CE$ and $BE = DE$).

  6. Each diagonal divides the parallelogram into two congruent triangles ($\Delta ABC \cong \Delta ADC$).

• Five Conditions to Prove a Quadrilateral is a Parallelogram

  1. Both pairs of opposite sides are parallel and congruent.

  • Example: If $\overline{NE} \parallel \overline{IC}$ and $\overline{NI} \cong \overline{EC}$, then $\square NICE$ is a parallelogram.

  1. Both pairs of opposite angles are congruent.

  • Example: If $\angle L \cong \angle M = 110^\circ$ and $\angle A \cong \angle K = 70^\circ$, then $\square MATH$ is a parallelogram.

  1. Any pair of consecutive angles is supplementary.

  • Example: If $m\angle M + m\angle A = 180^\circ$, then $\square MAKE$ is a parallelogram.

  1. Diagonals bisect each other.

  • Example: If $AX = CX$ and $DX = BX$, then $\square ABCD$ is a parallelogram.

  1. A diagonal divides the quadrilateral into two congruent triangles.

  • Example: If $\Delta ABC \cong \Delta ADC$, then $\square ABCD$ is a parallelogram.

• Parallelogram Area Formulas

  • Area: $A = b \times h$

  • Base: $b = \frac{A}{h}$

  • Height: $h = \frac{A}{b}$

  • Example Problem 1: Given $A = 144\,\text{m}^2$ and $b = 16\,\text{m}$.     $h = \frac{144\,\text{m}^2}{16\,\text{m}} = 9\,\text{m}$

  • Example Problem 2: Given $A = 240\,\text{m}^2$ and $h = 15\,\text{m}$.     $b = \frac{240\,\text{m}^2}{15\,\text{m}} = 16\,\text{m}$

Special Parallelograms: Rectangles, Rhombuses, and Squares

• Rectangle

  • Definition: A parallelogram with four right angles ($90^\circ$).

  • Specific Properties:

  1. All angles are equal right angles: $m\angle W = m\angle X = m\angle Y = m\angle Z = 90^\circ$.

  2. Diagonals are congruent: $\overline{XZ} \cong \overline{WY}$.

  3. Diagonals divide the rectangle into two pairs of congruent triangles.

• Rhombus

  • Definition: A parallelogram with four congruent sides.

  • Specific Properties:

  1. All sides are congruent: $MN = NO = OP = PM$.

  2. Diagonals intersect perpendicularly to form four $90^\circ$ angles ($\angle MRN = \angle NRO = \angle ORP = \angle PRM = 90^\circ$).

  3. Diagonals bisect the interior angles of the rhombus.

  4. Diagonals divide the rhombus into four congruent right/isosceles triangles ($\Delta QMR \cong \Delta RMS \cong \Delta SMT \cong \Delta TMQ$).

• Square

  • Definition: A parallelogram with four congruent sides and four right angles.

  • Specific Properties:

  1. Combines all properties of both a rectangle and a rhombus.

  2. All angles equal $90^\circ$ ($m\angle Q = m\angle R = m\angle S = m\angle T = 90^\circ$).

  3. Diagonals intersect at $90^\circ$ and bisect the corner angles into two $45^\circ$ angles ($\angle TQM = \angle RQM = \angle QRM = \angle SRM = 45^\circ$).

Algebraic Applications and Worked Examples

• Example 1: Diagonals and Consecutive Angles

  • Solving for $y$ using bisected diagonals:     $2y - 10 = y - 8$     $2y - y = 10 - 8$     $y = 2$

  • Solving for $x$ using bisected diagonals:     $3x + 6 = 12$     $3x = 12 - 6$     $3x = 6$     $x = 2$

  • Solving for $x$ using opposite angles $(x + 60)^\circ$ and $(3x + 40)^\circ$:     $x + 60 = 3x + 40$     $60 - 40 = 3x - x$     $20 = 2x$     $x = 10$

  • Solving for $y$ using supplementary consecutive angle relationship ($10 + 5y = 180$):     $10 + 5y = 180$     $5y = 180 - 10$     $5y = 170$     $y = 34$

  • Final Solutions: $x = 10$, $y = 34$

• Example 2: Opposite Sides

  • Solving for $x$:     $5x - 25 = 2x + 5$     $5x - 2x = 25 + 5$     $3x = 30$     $x = 10$

  • Solving for $y$:     $y + 18 = 3y$     $y - 3y = -18$     $-2y = -18$     $y = 9$

  • Final Solutions: $x = 10$, $y = 9$

• Example 3: Opposite Sides

  • Solving for $x$:     $5x + 2 = 6x - 3$     $5x - 6x = -2 - 3$     $-x = -5$     $x = 5$

  • Solving for $y$:     $5y - 6 = 7y - 12$     $5y - 7y = 6 - 12$     $-2y = -6$     $y = 3$

  • Final Solutions: $x = 5$, $y = 9$ (or $y = 3$ depending on constant pairing)

• Example 4: Opposite Angles

  • Solving for $n$:     $3n - 17 = 2n + 24$     $3n - 2n = 17 + 24$     $n = 41$

  • Solving for $m$:     $5m - 6 = m + 58$     $5m - m = 6 + 58$     $4m = 64$     $m = 16$

  • Final Solutions: $n = 41$, $m = 16$

• Example 5: Angle Calculations in Rectangle $MNOP$

  • Given: $\angle 1 = 37^\circ$

  • Find $\angle 2$:     $\angle 1 + \angle 2 = 90^\circ$     $37^\circ + \angle 2 = 90^\circ$     $\angle 2 = 90^\circ - 37^\circ = 53^\circ$ (noted as $43^\circ$ in transcription steps)

  • Find $\angle 11$:     $37^\circ + 37^\circ + \angle 11 = 180^\circ$     $74^\circ + \angle 11 = 180^\circ$     $\angle 11 = 180^\circ - 74^\circ = 106^\circ$

  • Find $\angle 12$:     $43^\circ + 43^\circ + \angle 12 = 180^\circ$     $86^\circ + \angle 12 = 180^\circ$     $\angle 12 = 180^\circ - 86^\circ = 94^\circ$

Formal Two-Column Proofs of Parallelogram Theorems

• Proof 1: Opposite Sides of a Parallelogram are Congruent

  • Given: Parallelogram $LOVE$ with diagonal $\overline{LV}$

  • Prove: $\overline{LO} \cong \overline{VE}$ and $\overline{OV} \cong \overline{LE}$

  • Proof Structure:

  1. Parallelogram $LOVE$ | Given

  2. $\overline{LO} \parallel \overline{VE}$ and $\overline{OV} \parallel \overline{LE}$ | Definition of a parallelogram

  3. Draw $\overline{LV}$ | Line Postulate

  4. $\angle LOV \cong \angle VEL$ and $\angle LEV \cong \angle VOE$ | If two parallel lines are cut by a transversal, then alternate interior angles are congruent

  5. $\overline{LV} \cong \overline{LV}$ | Reflexive Property

  6. $\Delta LOV \cong \Delta VEL$ | ASA Congruence Postulate

  7. $\overline{LO} \cong \overline{VE}$ and $\overline{OV} \cong \overline{LE}$ | Corresponding parts of congruent triangles are congruent (CPCTC)

• Proof 2: Consecutive Angles of a Parallelogram are Supplementary

  • Given: Parallelogram $LOVE$

  • Prove: $\angle L$ and $\angle O$ are supplementary; $\angle L$ and $\angle E$ are supplementary; $\angle O$ and $\angle V$ are supplementary; $\angle V$ and $\angle E$ are supplementary

  • Proof Structure:

  1. Parallelogram $LOVE$ | Given

  2. $\overline{LO} \parallel \overline{VE}$ and $\overline{OV} \parallel \overline{LE}$ | Definition of a parallelogram

  3. $\angle L$ and $\angle O$ are supplementary | If two parallel lines are cut by a transversal, then same-side interior angles are supplementary

  4. $\angle L \cong \angle V$ and $\angle O \cong \angle E$ | Opposite angles of a parallelogram are congruent

  5. $\angle V$ and $\angle E$ are supplementary | An angle that is supplementary to one of two congruent angles is supplementary to the other

• Proof 3: The Diagonals of a Parallelogram Bisect Each Other

  • Given: Parallelogram $LOVE$ with diagonals $\overline{LV}$ and $\overline{OE}$ intersecting at $S$

  • Prove: $\overline{LS} \cong \overline{VS}$ and $\overline{ES} \cong \overline{OS}$

  • Proof Structure:

  1. Parallelogram $LOVE$ with diagonals $\overline{LV}$ and $\overline{OE}$ | Given

  2. $\overline{LO} \cong \overline{VE}$ | Opposite sides of a parallelogram are congruent

  3. $\overline{LO} \parallel \overline{VE}$ | Definition of a parallelogram

  4. $\angle 2 \cong \angle 3$ and $\angle 1 \cong \angle 4$ | If two parallel lines are cut by a transversal, then alternate interior angles are congruent

  5. $\Delta LSO \cong \Delta VSE$ | ASA Congruence Postulate

  6. $\overline{LS} \cong \overline{VS}$ and $\overline{ES} \cong \overline{OS}$ | Corresponding parts of congruent triangles are congruent (CPCTC)

• Proof 4: Each Diagonal Division into Congruent Triangles

  • Given: Parallelogram $LOVE$ with diagonal $\overline{LV}$

  • Prove: $\Delta LEV \cong \Delta OVL$

  • Proof Structure:

  1. Parallelogram $LOVE$ with diagonal $\overline{LV}$ | Given

  2. $\overline{LO} \parallel \overline{VE}$ and $\overline{OV} \parallel \overline{LE}$ | Definition of a parallelogram

  3. $\angle LVO \cong \angle OVL$ | If two parallel lines are cut by a transversal, then alternate interior angles are congruent

  4. $\overline{LV} \cong \overline{LV}$ | Reflexive Property

  5. $\Delta LEV \cong \Delta OVL$ | If two parallel lines are cut by a transversal, then alternate interior angles are congruent / ASA Congruence Postulate