Math 112Z Lecture 13 - Detailed Study Notes

Math 112Z Lecture 13

Triangle Geometry
  • Triangles: A fundamental shape in geometry defined by three sides and three angles.
Similar Triangles
  • Definition of Similar Triangles: Two triangles are similar if they can be transformed into one another by scaling (resizing), possibly involving rotation and reflection.
  • Characteristics of Similar Triangles:
    • They have the same angle measures.
    • Their corresponding sides are in proportion to each other.
Congruent Triangles
  • Definition of Congruent Triangles: Two triangles are congruent if they have the exact same shape and size, which means all their corresponding sides and angles are equal.
  • Characteristics of Congruent Triangles:
    • They have the same side lengths.
    • They have the same angle measures.
Criteria for Triangle Similarity and Congruence
  • Similarity Condition:
    • Triangles are similar if they have:
    • Same angle measures.
  • Congruence Condition:
    • Triangles are congruent if they have:
    • Same side lengths.
  • Additional Conditions for Congruence:
    • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
    • AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, the triangles are congruent.
    • ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
  • Non-valid Congruence Condition:
    • SSA (Side-Side-Angle) is not a valid criterion for establishing congruence as it may lead to ambiguous cases.
Goal of Triangle Study
  • Objective: Given sufficient information about a triangle, determine all relevant information (find all unknown sides and angles).
Example Problem
  • Given two side lengths (denoted as a and b) and the angle (denoted as O) between them, we can find the third side length (denoted as c).
New Area Formula for Triangles

  • Area Calculation:

    • The area of triangle ABC can be calculated using the formula:

    Area[△ABC]=12bh\text{Area}[\triangle ABC] = \frac{1}{2}bh

    where b is the base and h is the height.

  • Using Sine for Area:

    • A new area formula derived from sine of the angle:

    Area[△ABC]=12absin(C)\text{Area}[\triangle ABC] = \frac{1}{2}absin(C)

    where a and b are the lengths of two sides and C is the included angle.

Notation for Angles and Sides
  • Angle Notation:
    • Angles in the triangle are labeled as A, B, C. Side lengths opposite these angles are labeled as a, b, c respectively.
    • In some instances, lower-case letters are used for sides, and uppercase for angles.
  • Summary of Notation:
    • A: Angle at vertex A
    • B: Angle at vertex B
    • C: Angle at vertex C
    • a: Side opposite angle A
    • b: Side opposite angle B
    • c: Side opposite angle C