6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles

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1 Chapter 1: Guide Scope, AFQT & Service-Specific Policy

1.1 Editorial Scope and Service-Specific AFCT Policy 1.2 AFQT Meaning and Study Priorities 1.3 Service Composites and Targeted Skill Planning 1.4 Format-Agnostic Test Preparation and Pacing

2 Chapter 2: General Science (GS)

2.1 Life Science: Cell Biology, Human Anatomy & Genetics 2.2 Physical Science: Matter, Energy, Forces & Motion 2.3 Earth & Space Science: Geology, Meteorology & Astronomy 2.4 Chemistry Fundamentals: Atomic Structure, Elements & Chemical Reactions

3 Chapter 3: Arithmetic Reasoning (AR)

3.1 Word Problems: Rates, Ratios, Proportions & Percentages 3.2 Financial & Distance Math: Interest, Profit/Loss, Speed & Time 3.3 Work & Mixture Problems: Joint Rates & Percentage Concentrations 3.4 Multi-Step Problem Solving & Algebraic Word Problems

4 Chapter 4: Word Knowledge (WK)

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5 Chapter 5: Paragraph Comprehension (PC)

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6 Chapter 6: Mathematics Knowledge (MK)

6.1 Pre-Algebra & Number Operations: Exponents, Radicals & Order of Operations 6.2 High School Algebra: Linear Equations, Inequalities & Polynomials 6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles 6.4 Coordinate Geometry, Perimeter, Area & Volume Formulas

7 Chapter 7: Electronics Information (EI)

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8 Chapter 8: Auto & Shop Information (AS)

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9 Chapter 9: Mechanical Comprehension (MC)

9.1 Simple Machines: Levers, Pulleys, Inclined Planes & Mechanical Advantage 9.2 Force, Motion, Work & Power: Newton's Laws & Energy Transformations 9.3 Fluid Dynamics & Pressure: Pascal's Principle, Hydraulics & Pneumatics 9.4 Mechanical Components: Gears, Belts, Springs & Friction

10 Chapter 10: Spatial Assembly Practice (AO Skill)

10.1 Spatial Connections and Assembly Patterns

6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles

Key Takeaways Explain This Topic AI

Distinguish angle relationships: complementary (sum to 90 degrees), supplementary (sum to 180 degrees), and vertical angles (equal).

Apply parallel line transversal theorems: corresponding, alternate interior, and alternate exterior angles are equal.

Master right triangle theorems: Pythagorean Theorem (a^2 + b^2 = c^2) and special right triangles (30-60-90 and 45-45-90).

Calculate interior angle sums for n-sided polygons using (n - 2) * 180 degrees.

Apply circle formulas for circumference (2 pi r), area (pi r^2), arc length ((theta/360) 2 pi r), and sector area ((theta/360) pi r^2).

Last updated: July 2026

6.3 Geometry Fundamentals: Angles, Triangles, Quadrilaterals & Circles

Geometry tests your ability to analyze geometric figures, compute angle measures, apply geometric theorems, and solve spatial relationships. in Mathematics Knowledge practice, geometry questions require a strong foundation in planar geometry properties, right triangle relationships, and circle definitions.

Angle Relationships & Lines

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

• Classification of Angles
• Acute Angle: 0 ∘ < θ < 90 ∘ 0^\circ < \theta < 90^\circ 0 ∘ < θ < 9 0 ∘
• Right Angle: θ = 90 ∘ \theta = 90^\circ θ = 9 0 ∘ (indicated by a square symbol ⌞ \llcorner └ )
• Obtuse Angle: 90 ∘ < θ < 180 ∘ 90^\circ < \theta < 180^\circ 9 0 ∘ < θ < 18 0 ∘
• Straight Angle: θ = 180 ∘ \theta = 180^\circ θ = 18 0 ∘ (forms a straight line)
• Key Angle Relationships

• Angle Pair Definition / Property Equation
• Complementary Angles Two angles whose sum is 90 ∘ 90^\circ 9 0 ∘ A + B = 90 ∘ A + B = 90^\circ A + B = 9 0 ∘

Supplementary Angles Two angles whose sum is 180 ∘ 180^\circ 18 0 ∘ A + B = 180 ∘ A + B = 180^\circ A + B = 18 0 ∘

• Linear Pair Adjacent angles that form a straight line A + B = 180 ∘ A + B = 180^\circ A + B = 18 0 ∘
• Vertical Angles Opposite angles formed by intersecting lines A = B A = B A = B (Equal)
• Parallel Lines Cut by a Transversal

When two parallel lines ( L 1 ∥ L 2 L_1 \parallel L_2 L 1 ​ ∥ L 2 ​ ) are intersected by a third line (transversal T T T ), eight angles are formed with unique relationships:

Equal Angles:

• Corresponding Angles: (e.g., Position 1 and Position 5)
• Alternate Interior Angles: (e.g., Position 3 and Position 6)
• Alternate Exterior Angles: (e.g., Position 1 and Position 8)
• Vertical Angles: (e.g., Position 1 and Position 4)

Supplementary Angles ( 180 ∘ 180^\circ 18 0 ∘ ):

• Consecutive Interior Angles (Same-side interior): A + B = 180 ∘ A + B = 180^\circ A + B = 18 0 ∘
• Consecutive Exterior Angles (Same-side exterior): A + B = 180 ∘ A + B = 180^\circ A + B = 18 0 ∘
• Triangles & Triangle Theorems

A triangle is a three-sided polygon. The sum of the interior angles of any triangle is always 180 ∘ 180^\circ 18 0 ∘ .

Classifications of Triangles

By Sides:

Scalene: All three sides have different lengths.

Isosceles: At least two sides are equal in length; angles opposite equal sides are equal.

Equilateral: All three sides are equal; all three angles equal 60 ∘ 60^\circ 6 0 ∘ .

By Angles:

Acute: All three angles are < 90 ∘ < 90^\circ < 9 0 ∘ .

Right: One angle equals 90 ∘ 90^\circ 9 0 ∘ .

Obtuse: One angle is > 90 ∘ > 90^\circ > 9 0 ∘ .

Core Triangle Theorems

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side ( a + b > c a + b > c a + b > c ).

Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles ( m ∠ Ext = m ∠ A + m ∠ B m\angle \text{Ext} = m\angle A + m\angle B m ∠ Ext = m ∠ A + m ∠ B ).

Right Triangles & Special Relationships

In any right triangle with legs a a a and b b b and hypotenuse c c c (the side opposite the 90 ∘ 90^\circ 9 0 ∘ angle): Pythagorean Theorem: a 2 + b 2 = c 2 \text{Pythagorean Theorem: } a^2 + b^2 = c^2 Pythagorean Theorem: a 2 + b 2 = c 2

Common Pythagorean Triples

Recognizing common integer side length ratios saves unnecessary calculation time in practice:

3 − 4 − 5 3 - 4 - 5 3 − 4 − 5 (and multiples: 6 − 8 − 10 , 9 − 12 − 15 , 12 − 16 − 20 6-8-10, 9-12-15, 12-16-20 6 − 8 − 10 , 9 − 12 − 15 , 12 − 16 − 20 )

• 5 − 12 − 13 5 - 12 - 13 5 − 12 − 13 (and multiples: 10 − 24 − 26 10-24-26 10 − 24 − 26 )
• 8 − 15 − 17 8 - 15 - 17 8 − 15 − 17
• 7 − 24 − 25 7 - 24 - 25 7 − 24 − 25
• Special Right Triangles

Practice questions may test two special right triangles derived from regular figures: 45°-45°-90° Triangle (Isosceles Right) 30°-60°-90° Triangle

• /| /|
• / | / |
• x√2 / | x 2x / | x√3
• / | / |
• /____| /____|
• x x

45 ∘ − 45 ∘ − 90 ∘ 45^\circ - 45^\circ - 90^\circ 4 5 ∘ − 4 5 ∘ − 9 0 ∘ Triangle (Isosceles Right Triangle):

Ratio of side lengths: 1 : 1 : 2 1 : 1 : \sqrt{2} 1 : 1 : 2 ​

Leg = x = x = x , Hypotenuse = x 2 = x\sqrt{2} = x 2 ​ .

30 ∘ − 60 ∘ − 90 ∘ 30^\circ - 60^\circ - 90^\circ 3 0 ∘ − 6 0 ∘ − 9 0 ∘ Triangle:

• Ratio of side lengths: 1 : 3 : 2 1 : \sqrt{3} : 2 1 : 3 ​ : 2
• Side opposite 30 ∘ 30^\circ 3 0 ∘ (Short Leg) = x = x = x
• Side opposite 60 ∘ 60^\circ 6 0 ∘ (Long Leg) = x 3 = x\sqrt{3} = x 3 ​
• Hypotenuse = 2 x = 2x = 2 x
• Quadrilaterals & Polygon Theorems

A polygon is a closed 2D shape with straight sides. An n n n -sided polygon has n n n interior angles.

• Polygon Formulas
• Sum of Interior Angles: S = ( n − 2 ) × 180 ∘ S = (n - 2) \times 180^\circ S = ( n − 2 ) × 18 0 ∘

Each Interior Angle (Regular Polygon): I = ( n − 2 ) × 180 ∘ n I = \frac{(n - 2) \times 180^\circ}{n} I = n ( n − 2 ) × 18 0 ∘ ​

Sum of Exterior Angles: Always 360 ∘ 360^\circ 36 0 ∘ for any convex polygon.

• Each Exterior Angle (Regular Polygon): E = 360 ∘ n E = \frac{360^\circ}{n} E = n 36 0 ∘ ​
• Quadrilateral Hierarchy

Parallelogram: Opposite sides parallel and equal; opposite angles equal; diagonals bisect each other.

Rectangle: Parallelogram with four 90 ∘ 90^\circ 9 0 ∘ right angles; diagonals are equal.

Rhombus: Parallelogram with four equal sides; diagonals are perpendicular bisectors.

Square: Regular quadrilateral (both rectangle and rhombus); four equal sides and four 90 ∘ 90^\circ 9 0 ∘ angles.

Trapezoid: Quadrilateral with exactly one pair of parallel bases.

Circle Fundamentals

A circle is the set of all points in a plane equidistant from a fixed point (center).

Key Definitions

Radius ( r r r ): Distance from center to any point on circle.

Diameter ( d d d ): Distance across circle through center ( d = 2 r d = 2r d = 2 r ).

Chord: Line segment connecting any two points on the circle.

Tangent: Line intersecting circle at exactly one point (perpendicular to radius at point of tangency).

• Essential Circle Formulas
• Circumference: C = 2 π r = π d C = 2\pi r = \pi d C = 2 π r = π d
• Area: A = π r 2 A = \pi r^2 A = π r 2
• Arc Length: L = θ 360 ∘ × 2 π r L = \frac{\theta}{360^\circ} \times 2\pi r L = 36 0 ∘ θ ​ × 2 π r

Sector Area: A sector = θ 360 ∘ × π r 2 A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2 A sector ​ = 36 0 ∘ θ ​ × π r 2