Triangle congruence is all about proving two triangles are identical in shape and size. We'll look at different ways to show triangles are congruent, like SSS, SAS, and ASA. These methods help us solve real-world problems involving triangles.

We'll also dive into equilateral triangles, which have three equal sides and angles. Understanding their special properties will make solving triangle problems easier. By the end, you'll be a pro at spotting and using triangle congruence in various situations.

Triangle Congruence

Congruence of overlapping triangles

• Congruence postulates and theorems prove triangles are congruent based on specific criteria
  • Side-Side-Side (SSS) Postulate: Three pairs of congruent sides guarantee triangle congruence
  • Side-Angle-Side (SAS) Postulate: Two pairs of congruent sides and the included angle prove triangle congruence
  • Angle-Side-Angle (ASA) Postulate: Two pairs of congruent angles and the included side establish triangle congruence
  • Angle-Angle-Side (AAS) Theorem: Two pairs of congruent angles and a non-included side confirm triangle congruence
  • Hypotenuse-Leg (HL) Theorem: In right triangles, a congruent hypotenuse and leg result in triangle congruence
• Identify overlapping triangles by recognizing shared sides or angles and using given information to determine congruent parts
• Prove congruence by selecting the appropriate postulate or theorem, stating the congruent parts, and concluding the triangles are congruent

Properties of equilateral triangles

• Equilateral triangles have three congruent sides and three congruent 60° angles
• Altitudes, angle bisectors, and perpendicular bisectors of sides coincide in equilateral triangles
• The centroid, incenter, and circumcenter are the same point in equilateral triangles
• Solve problems using the congruent sides to create equations and the 60° angles to find missing measurements
• Prove congruence of equilateral triangles using SSS or the special case of AAA congruence

Applying Triangle Congruence

Relationships in congruent triangles

• Congruent triangles have congruent corresponding sides and angles
• Overlapping triangles share congruent sides or angles in both triangles
• Equilateral triangles have side length ratios of 1:1:1 and angle measures that sum to 180°
• Use proven congruence to identify congruent parts and set up equations or ratios to solve for missing values

Problem-solving with triangle congruence

1. Identify the given information and the desired outcome by determining congruent or shared parts and the specific value or relationship to be found
2. Apply the appropriate congruence postulate or theorem to prove triangle congruence and use equilateral triangle properties when relevant
3. Set up equations using the congruent parts and relationships, solve for the desired value using algebra and geometry principles, and justify each step using the given information, congruence principles, and triangle properties