Applied Geometry 01/12/26 Section 4-3

Congruence in Triangles

Definition of Congruence

  • Congruent: Means same or equal.
    • Interpretations of 'same':
    • Same color.
    • Same shape.
    • Same weights.
    • Same size.
  • By definition, congruent shapes have the same size and shape.

Congruent Triangles

  • When discussing two triangles, if they are congruent:
    • They have equal measurements in terms of sides and angles.

Pairs of Congruent Sides

  • With respect to two congruent triangles:
    • Naming Congruent Sides:
    • Example pairs that might be named include:
      • AC is congruent to DF (using corresponding locations).
      • AB is congruent to DE.
      • BC is congruent to EF.
  • Important to note:
    • When identifying congruent parts, one must match corresponding positions.

Pairs of Congruent Angles

  • For angles in two congruent triangles:
    • Angle Correspondences:
    • Angle A is congruent to Angle D.
    • Angle B is congruent to Angle E.
    • Angle C is congruent to Angle F.
  • It is crucial that angle correspondences also maintain the same relative positions (e.g., A aligns with D, B aligns with E, etc.).

Total Parts of Congruent Triangles

  • Each pair of congruent triangles contains:
    • Total: 3 pairs of sides + 3 pairs of angles = 6 total parts.

Congruence Criteria for Triangles

  • Not all parts need to be known for congruence. Triangle congruences can be determined using specific rules:
Side-Side-Side (SSS) Congruence Rule
  • Definition: If all three pairs of sides of two triangles are congruent, then the triangles are congruent.
  • Key Point: Knowing the sides alone is sufficient to prove triangle congruence without needing to consider the angles.
Side-Angle-Side (SAS) Congruence Rule
  • Definition: If two sides of a triangle are congruent to two sides of another triangle, and the included angle (the angle between those two sides) is also congruent, then the triangles are congruent.
  • Key Point: The included angle must be the angle formed between the two known sides.
    • Illustration: If we denote triangle JKL, where JK and KL are the congruent sides, then angle K is regarded as the included angle.

Included Angles and Sides

  • Included Angle: The angle formed between two sides of a triangle.
    • Example: Between sides JK and KL, the included angle is Angle K.
  • Identifying Included Sides:
    • Ask for the included side between two specified angles (e.g., angle E and angle F).
    • The correct included side for angle F and angle E is FB (the side between these two angles).

Important Reminders

  • Distinction between Angles and Sides: It is essential to differentiate clearly between sides and angles when naming them.
    • Incorrect identification can lead to misunderstanding (e.g., naming a side instead of an angle).
  • Encouragement: If any confusion arises during problem-solving, students are urged to seek clarification.