Basic Geometry Concepts: Angles, Lines, Triangles

Geometry Fundamentals

Understanding Angles

An angle is the portion of a plane formed by two rays (semi-straight lines) sharing a common endpoint. The rays are called the sides of the angle, and the common endpoint is called the vertex.

Angle Designation

• By three letters, with the vertex letter always in the middle (e.g., ∠ABC).
• By the letter of the vertex (e.g., ∠B).
• By a number or a Greek letter, often placed near the vertex (e.g., ∠α).

Types of Angles

• Adjacent angles: Angles that share a common vertex and a common side.
• Linear Pair: Two adjacent angles whose non-common sides form a straight line (sum is 180°).
• Right angle: An angle that measures exactly 90°.
• Straight angle: An angle that measures exactly 180°.
• Complementary angles: Two angles whose measures sum to 90° (they do not have to be adjacent).
• Supplementary angles: Two angles whose measures sum to 180° (they do not have to be adjacent).
• Vertical angles: Non-adjacent angles formed by two intersecting lines. Property: Vertical angles are equal in measure.
• Acute angle: An angle whose measure is less than 90°.
• Obtuse angle: An angle whose measure is greater than 90° but less than 180°.
• Reflex angle: An angle whose measure is greater than 180° but less than 360°.
• Full angle: An angle that measures exactly 360°.
• Angle of rotation: An angle whose measure is greater than 360° (representing multiple rotations).

Angle Bisector

An angle bisector is a ray that divides an angle into two equal angles.

Lines in Geometry

• Secant lines: Lines that intersect at exactly one point.
• Perpendicular lines: Lines that intersect to form four right angles (90° angles).
• Parallel lines: Lines in the same plane that never intersect.

Line Segments

• Line segment: A portion of a line that is bounded by two distinct endpoints. These points are the endpoints of the segment.
• Segment bisector: A line, ray, or segment that passes through the midpoint of a segment. A perpendicular bisector is a line, ray, or segment that is perpendicular to a segment and passes through its midpoint.

Introduction to Triangles

A triangle is a polygon with three edges and three vertices. It is a portion of the plane bounded by three line segments. The points where the segments intersect are called the vertices (e.g., A, B, C). The segments connecting the vertices are called the sides (e.g., AB, BC, CA). The angles formed inside the triangle by the sides are called the internal angles.

Triangle Denotation

• Vertices are denoted by capital letters (e.g., A, B, C).
• Sides are denoted by lowercase letters corresponding to the opposite vertex (e.g., side 'a' is opposite vertex A).
• Angles are often denoted by Greek letters (e.g., α, β, γ) or by the vertex letter (e.g., ∠A).

External Angles

External angles of a triangle are formed by extending one side of the triangle. An external angle is adjacent to an internal angle and supplementary to it.