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3: Geometry

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    174257

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    • 3.1: Types of Angles and Degree Measure
      This page defines fundamental geometric concepts related to angles, including types (acute, obtuse, reflex, right) and their measures in degrees. It presents the Angle Addition Postulate, complementary and supplementary angles, and includes examples for classifying angles and performing arithmetic with them. Notation conventions for clear angle representation are also discussed.
    • 3.2: Vertical, Alternate, and Corresponding Angles
      This page explores the relationships between angles created by intersecting and parallel lines, defining key terms like vertical angles and transversals. It covers important theorems, such as the Congruency of Vertical Angles and Alternate Angles Theorem, that outline conditions for angle congruence and parallelism. Examples are provided to demonstrate the application of these concepts and theorems in solving for unknown angles.
    • 3.3: Triangles
      This page covers the essential properties of triangles, including their definitions, classifications (equilateral, isosceles, scalene, right, acute, and obtuse), and key theorems such as the Triangle Sum Theorem and the Isosceles Triangle Theorem. It explains altitudes and classification based on side lengths and angles, using examples to illustrate triangle classification and finding missing angles, highlighting the significance of these properties in geometry.
    • 3.4: The Triangle Inequality Theorem
      This page covers important theorems related to triangles, such as Angle-Side Correspondence and the Triangle Inequality and its converse. It clarifies that the longer side is opposite the larger angle and that the sum of any two sides must be greater than the third. The page includes examples of valid triangles and degenerate cases, emphasizing conditions for forming triangles. Overall, it provides a foundational understanding of triangle properties and inequalities.
    • 3.5: The Pythagorean Theorem, Distance Formula, and Midpoint Formula
      This page discusses the Pythagorean Theorem and its relevance in coordinate geometry, detailing the relationship \(a^2 + b^2 = c^2\) for right triangles. It explains Pythagorean triples and introduces the distance and midpoint formulas. The content is reinforced through examples that demonstrate the practical application of these mathematical concepts.
    • 3.6: Similar Triangles
      This page covers fundamental concepts in triangle geometry, focusing on congruency and similarity. It explains that congruent triangles have equal angles and sides, while similar triangles have congruent angles and proportional sides determined by a scale factor. Theorems such as AA, SSS, and SAS are introduced for assessing similarity, accompanied by examples that illustrate their use in solving for missing side lengths and proportions.
    • 3.7: Special Triangles
      This page explains special right triangles, specifically 30-60-90 and 45-45-90 triangles, outlining their definitions and relationships between sides. In a 30-60-90 triangle, the hypotenuse is double the shortest side, and the longer leg is \( \sqrt{3} \) times the shortest side. In a 45-45-90 triangle, both legs are equal, and the hypotenuse is \( \sqrt{2} \) times a leg.
    • 3.8: Circles - Properties, Area, and Circumference
      This page introduces essential definitions and theorems about circles in geometry, detailing the concepts of radius, diameter, circumference, and area. It defines a circle as the set of points equidistant from a center and presents mathematical expressions for circumference and area. The text highlights the importance of accurately using radius and diameter in calculations, provides examples for determining these measurements, and emphasizes preserving results in terms of π for precision.
    • 3.9: Area and Perimeter of Triangles and Quadrilaterals
      This page explains the definitions and theorems for measuring perimeter and area of common geometric figures like polygons, triangles, and trapezoids. It defines perimeter as the total length of a polygon's sides and area as the number of unit squares within a region. The importance of using perpendicular height in calculations is highlighted to prevent errors. Additionally, it provides formulas for area and perimeter for various shapes, along with examples to illustrate their application.
    • 3.10: Volumes and Surface Areas of Common Shapes
      This page covers key concepts of volume and surface area of three-dimensional solids, defining volume as the space an object occupies and surface area as its exterior area. It outlines essential formulas for prisms, cylinders, spheres, and cones, including lateral surface area and slant height. It also includes a specific example calculating the radius, surface area, and volume of a sphere with a volume of 27 cm³.

    This page titled 3: Geometry is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Roy Simpson, Cosumnes River College.