Triangle Geometry Overview

Overview

This lecture introduces triangles, covering their definitions, types, important theorems, and properties, along with key formulas and concepts in triangle geometry.

Triangle Basics

• A triangle has three sides, three angles, and three vertices (corners).
• The sum of the interior angles in any triangle is always 180°.

Types of Triangles

• By sides:
  • Equilateral: three equal sides, all angles 60°.
  • Isosceles: two equal sides, angles opposite equal sides are equal.
  • Scalene: all sides and angles are different.
• By angles:
  • Acute: all angles less than 90°.
  • Right: one angle exactly 90°.
  • Obtuse: one angle greater than 90°.

Triangle Theorems and Properties

• Exterior Angle Theorem: An exterior angle equals the sum of the two non-adjacent interior angles.
• Triangle Inequality Theorem: The sum of any two sides must be greater than the third side.
• Perimeter: The sum of all three side lengths.
• Base and Height: Any side can be the base; the height is a perpendicular line from the opposite vertex to the base.

Area and Key Formulas

• Area formula: (1/2) × base × height.
• In right triangles, one leg can be the height if it meets the base at 90°.
• In obtuse triangles, the height may fall outside the triangle.

Congruence and Similarity

• Congruent triangles: Same shape and size; checked by SSS, SAS, ASA, AAS, or RHS criteria.
• Similar triangles: Same shape, different size; checked by AA, SSS (ratio), or SAS (ratio and included angle).

Special Theorems

• Pythagorean Theorem: In right triangles, a² + b² = c² (c is hypotenuse).
• Median Theorem: A median connects a vertex to the midpoint of the opposite side and divides the area in half.
• Centroid Theorem: The three medians intersect at the centroid, which divides each median in a 2:1 ratio and is the triangle’s balance point.
• Angle Bisector Theorem: An angle bisector divides the opposite side in the same ratio as the other two sides.

Equilateral Triangle Properties

• All sides and angles are equal.
• Height, median, and angle bisector from each vertex are the same line.

Key Terms & Definitions

• Triangle — a three-sided polygon with three angles and three vertices.
• Vertex (plural: vertices) — a corner point of a triangle.
• Median — a line from a vertex to the midpoint of the opposite side.
• Centroid — the intersection point of all medians.
• Bisector — a line that divides an angle into two equal parts.
• Congruence — triangles that are identical in shape and size.
• Similarity — triangles with the same shape but different sizes.

Action Items / Next Steps

• Practice identifying and classifying triangles by sides and angles.
• Solve problems using the angle sum, perimeter, and area formulas.
• Attempt exercises on congruence and similarity criteria.
• Review the centroids and medians in various triangles.
• Prepare for part two of the lecture.