Summary of Plane geometry: triangle and circle

## Introduction to Plane Geometry Plane geometry is the branch of mathematics that studies two-dimensional shapes and figures. These shapes, like points, lines, angles, and triangles, exist on a flat surface, known as a plane. Understanding plane geometry is crucial because it forms the bedrock for more advanced mathematical concepts and has countless practical applications in fields such as architecture, engineering, and computer graphics. This material will guide you through the fundamental properties and theorems related to the basic elements of plane geometry and, specifically, triangles. ### Basic Geometric Elements To begin our journey into plane geometry, let's define its most fundamental building blocks: > **Point**: A point is a location in space that has no dimension (no length, width, or height). It's typically represented by a dot and labeled with a capital letter, like point $A$. > **Line**: A line is a one-dimensional figure that has no thickness and extends infinitely in two opposite directions. It is represented by a straight path with arrows on both ends, often denoted by two points on the line, e.g., line $AB$, or by a single lowercase letter, e.g., line $l$. > **Plane**: A plane is a two-dimensional flat surface that extends infinitely in all directions. Think of it as a perfectly flat sheet of paper that goes on forever. It's often represented by a parallelogram and labeled with a capital letter, e.g., plane $P$. > **Angle**: An angle is formed by two rays that share a common endpoint. This common endpoint is called the **vertex** of the angle, and the two rays are its **sides**. Angles are typically measured in degrees ($^{\circ}$). For example, an angle might be $45^{\circ}$ or $90^{\circ}$. ### Triangles A **triangle** is the simplest polygon and a foundational shape in geometry. It is a closed two-dimensional figure made up of three straight line segments (sides) and three vertices (where the sides meet). Triangles are everywhere, from the architecture of pyramids to the trusses in a bridge. Triangles can be classified in two primary ways: by the lengths of their sides and by the measures of their angles. #### Classification by Side Length: * **Equilateral triangle**: An equilateral triangle has all three of its sides equal in length. Consequently, all three of its interior angles are also equal, each measuring $60^{\circ}$. * **Isosceles triangle**: An isosceles triangle has at least two sides of equal length. The angles opposite these equal sides (called base angles) are also equal in measure. A special case of an isosceles triangle is the equilateral triangle. * **Scalene triangle**: A scalene triangle has all three of its sides of different lengths. As a result, all three of its interior angles also have different measures. #### Classification by Angle Measure: * **Right triangle**: A right triangle is defined by having exactly one interior angle that measures $90^{\circ}$. This angle is called the right angle. The side opposite the right angle is the longest side and is called the **hypotenuse**. * **Acute triangle**: An acute triangle is a triangle where all three of its interior angles are **acute**, meaning each angle measures less than $90^{\circ}$. For example, a triangle with angles $60^{\circ}$, $70^{\circ}$, and $50^{\circ}$ is an acute triangle. * **Obtuse triangle**: An obtuse triangle has exactly one interior angle that is **obtuse**, meaning it measures greater than $90^{\circ}$ but less than $180^{\circ}$. The other two angles in an obtuse triangle must be acute. Did you know that the word "triangle" comes from the Latin words "tri" (meaning three) and "angulus" (meaning corner or angle), literally translating to "three-cornered"? ## Fundamental Theorems of Triangles These theorems provide the rules that govern the behavior and properties of triangles. ### 1. Angle Sum Property of a Triangle **Theorem**: The sum of the interior angles of any triangle is