Basic Concepts in Geometry and Algebra

Basic Concepts in Geometry and Algebra

• Points identify specific locations on a plane, represented by capital letters, and have no size.
• Line Segments connect two endpoints, having a definite length, denoted by the endpoints.
• A Ray or Vector starts at a point and extends infinitely in one direction, characterized by an initial point and an arrowhead.

Perpendicular Lines and Angles

• Perpendicular Lines intersect to create four equal angles, forming a right angle (90º).
• An Angle is created when two rays or line segments meet at a point, known as the vertex.

Types of Angles

• A Straight Angle measures exactly 180º, with its rays forming a straight line.
• Acute Angles measure less than 90º, indicating sharpness.
• Obtuse Angles range from greater than 90º to less than 180º.
• Reflex Angles exceed 180º but fall below 360º (e.g., 270º).

Triangles

• A Triangle is a closed planar figure made of three line segments, having three sides and three angles; the angles always sum to 180º.

Types of Triangles

• A Right Triangle includes one 90º angle, with two legs forming the right angle and the third side, the hypotenuse.
• An Isosceles Triangle has two equal sides, and the angles opposite those sides are also equal.
• An Isosceles Right Triangle combines traits of both, with two equal sides and a right angle.
• An Equilateral Triangle features three equal sides and corresponding angles that are also equal.

Angle Relationships

• An Exterior Angle is equal to the sum of the two opposite interior angles in a triangle.
• Complementary Angles add up to 90º.
• Supplementary Angles total 180º.

Triangle Theorems

• SSS (Side, Side, Side): If three sides of one triangle match three sides of another, the triangles are congruent.
• SAS (Side, Angle, Side): Two sides and the included angle of one triangle equal those of another triangle.
• ASA (Angle, Side, Angle): Two angles and the included side correspond, resulting in congruency.
• AAS (Angle, Angle, Side): If two angles and a non-included side of one triangle match the same parts of another triangle, they are congruent.
• RHS (Right angle, Hypotenuse side): This theorem applies to right triangles to establish congruence.

Special Triangle Properties

• Legs are the two shorter sides that meet at the right angle in a right triangle.
• The Hypotenuse is the longest side, located opposite the right angle.

Pythagorean Theorem

• This theorem states that in a right triangle, the sum of the squares of the legs equals the square of the hypotenuse (a² + b² = c²).