4.1 Lines, Angles & Polygons

4.1 Lines, Angles & Polygons

Geometry on the GRE Quantitative Reasoning section tests your knowledge of fundamental geometric properties, geometric figures, and spatial reasoning. Rather than requiring formal two-column axiomatic proofs, the GRE assesses your ability to apply geometric definitions, theorems, and algebraic relationships to calculate angle measures, lengths, areas, and missing dimensions quickly and accurately.

Understanding basic primitive elements—points, lines, line segments, and rays—forms the foundation of geometric problem-solving on the GRE.

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1. Lines, Rays, and Basic Angle Definitions

• Line: A straight one-dimensional figure extending infinitely in two opposite directions.
• Line Segment: A part of a line bounded by two distinct end points. The distance between points $A$ and $B$ is denoted as $AB$.
• Ray: A part of a line that starts at an endpoint and extends infinitely in one direction.
• Parallel Lines: Two or more lines in the same plane that never intersect, regardless of how far they are extended. Parallel lines are denoted by the symbol $\parallel$ (e.g., $l_1 \parallel l_2$). Parallel lines have equal slopes in coordinate geometry.
• Perpendicular Lines: Two lines that intersect at a right angle ($90^\circ$). Perpendicularity is denoted by the symbol $\perp$ (e.g., $l_1 \perp l_2$). The slopes of two non-vertical perpendicular lines are negative reciprocals of each other ($m_1 \cdot m_2 = -1$).

Classification of Angles by Measure

An angle is formed by two rays sharing a common endpoint called the vertex. Angles are measured in degrees ($^\circ$):

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2. Angle Pair Relationships

Several core angle relationships appear frequently on the GRE and allow you to set up quick algebraic equations:

1. Complementary Angles: Two angles whose measures sum to $90^\circ$. $\theta_1 + \theta_2 = 90^\circ$ Example: If Angle $A = 35^\circ$, its complement is $90^\circ - 35^\circ = 55^\circ$.

2. Supplementary Angles: Two angles whose measures sum to $180^\circ$. $\theta_1 + \theta_2 = 180^\circ$ Angles that lie along a straight line (a linear pair) are always supplementary. Example: If Angle $B = 112^\circ$, its supplement is $180^\circ - 112^\circ = 68^\circ$.

3. Vertical Angles: Opposite angles formed by the intersection of two straight lines.

   • Key Property: Vertical angles are always equal in measure.
   • If line $AB$ intersects line $CD$ at point $P$, then $\angle APC = \angle BPD$ and $\angle APD = \angle BPC$.

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3. Parallel Lines Cut by a Transversal

When two parallel lines $l_1$ and $l_2$ are intersected by a third line called a transversal ($t$), eight distinct angles are formed. These eight angles break down into exactly two sets of four equal angles: four acute angles and four obtuse angles (assuming the transversal is non-perpendicular).

Categorization of Transversal Angles

GRE Tip (Big/Small Rule): When two parallel lines are cut by a transversal, all acute angles formed are equal to each other, all obtuse angles formed are equal to each other, and any acute angle plus any obtuse angle equals $180^\circ$.

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4. Polygons: Fundamentals, Interior & Exterior Angles

A polygon is a closed, two-dimensional geometric figure formed by three or more straight line segments (sides).

• Convex Polygon: A polygon in which all interior angles are strictly less than $180^\circ$, and no line segment connecting any two interior points passes outside the polygon. All polygons on the GRE are assumed to be convex unless explicitly stated otherwise.
• Concave Polygon: A polygon with at least one interior angle greater than $180^\circ$ (a reflex angle), causing part of the polygon to "cave in."

Interior Angle Sum Formula

For any convex polygon with $n$ sides (an $n$-gon), the sum of the interior angles $S$ is given by:

$S = (n - 2) \times 180^\circ$

Why does this formula work? Any $n$-sided polygon can be partitioned into $(n - 2)$ non-overlapping triangles by drawing diagonals from a single vertex. Since each triangle contains $180^\circ$, the total interior angle sum is $(n - 2) \times 180^\circ$.

Exterior Angle Sum Formula

An exterior angle is formed by extending one side of a polygon past its vertex. For any convex polygon, regardless of the number of sides $n$:

$\text{Sum of Exterior Angles} = 360^\circ$

Regular Polygons

A regular polygon is a polygon that is both equilateral (all sides equal in length) and equiangular (all interior angles equal in measure).

For a regular $n$-gon:

• Each Interior Angle: $\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$
• Each Exterior Angle: $\text{Exterior Angle} = \frac{360^\circ}{n}$
• Relationship: $\text{Interior Angle} + \text{Exterior Angle} = 180^\circ$.

Table of Polygon Angle Properties

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5. Worked GRE Step-by-Step Examples

Example 1: Parallel Line Transversal Algebra

Problem: Lines $L_1$ and $L_2$ are parallel. A transversal intersects $L_1$ and $L_2$. One interior angle is given as $(4x + 15)^\circ$ and its consecutive interior angle on the same side of the transversal is given as $(2x + 45)^\circ$. Find the value of $x$ and the measure of both angles.

Solution:

1. Identify the Angle Relationship: Consecutive interior angles formed by parallel lines cut by a transversal are supplementary. Their measures sum to $180^\circ$.
2. Set up the Equation: $(4x + 15) + (2x + 45) = 180$
3. Combine Like Terms: $6x + 60 = 180$
4. Solve for $x$: $6x = 120 \implies x = 20$
5. Calculate the Angle Measures:
   • First angle: $4(20) + 15 = 80 + 15 = 95^\circ$
   • Second angle: $2(20) + 45 = 40 + 45 = 85^\circ$
   • Check: $95^\circ + 85^\circ = 180^\circ$.

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Example 2: Regular Polygon Determination

Problem: Each interior angle of a regular polygon is five times the measure of each exterior angle. How many sides does this regular polygon have?

Solution:

1. Let $E$ be the exterior angle measure: $\text{Interior Angle } I = 5E$
2. Use the Linear Pair Property: $I + E = 180^\circ \implies 5E + E = 180^\circ \implies 6E = 180^\circ \implies E = 30^\circ$
3. Use the Exterior Angle Formula to find $n$: $E = \frac{360^\circ}{n} \implies 30^\circ = \frac{360^\circ}{n} \implies n = \frac{360}{30} = 12$
4. Conclusion: The polygon is a regular 12-gon (dodecagon) with 12 sides.

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6. GRE Exam Strategies & Common Pitfalls

• Figures are NOT Drawn to Scale: On the GRE Quantitative Reasoning section, geometric diagrams are generally not drawn to scale unless specifically noted. Do not estimate angle sizes or line segment lengths visually! Always rely strictly on stated geometric relationships, angle properties, and explicit values.
• Draw Auxiliary Parallel Lines: When faced with zig-zagging lines between parallel bounds (such as a "M-shaped" or "lightning bolt" angle problem), draw an additional parallel line through the vertex of the angle to split it into two manageable transversal problems.
• Exterior Angle Shortcut: Remember that finding the exterior angle ($360^\circ / n$) is almost always faster than working with the interior angle formula. If a problem asks for the number of sides $n$, calculate the exterior angle first!