ASA, SSS & SAS Triangle Postulates Study Notes

ASA, SSS & SAS Triangle Postulates | Properties & Examples

Contributors

  • Joao Amadeu

  • Jeff Calareso

Lesson Overview

This lesson introduces the concept of congruency applied to triangles through the ASA, SSS, and SAS triangle postulates.

Table of Contents

  • Congruent Triangles

  • Triangle Congruence Postulates or Theorems

  • ASA Geometry or ASA Postulate

  • Side-Angle-Side Theorem: SAS Postulate

  • Side-Side-Side Theorem: SSS Postulate

  • Examples and Applications

  • Comparison and Summary

Congruent Triangles

  • Congruence: In geometry, congruence refers to two figures having equal shapes and sizes.

  • Congruent Triangles: Two triangles are congruent if:

    • All three sides have the same length.

    • All three angles have the same measure.

  • Similarity vs Congruence:

    • Similar Triangles: Triangles are similar if their corresponding three angles are equal, but their sides do not need to be the same length.

    • Congruent triangles are always similar, but similar triangles are not necessarily congruent.

  • Transformation Effects: Transformations such as rotation, translation, and reflection will not affect the congruence of triangles.

Triangle Congruence Postulates or Theorems

To determine if two triangles are congruent, there are three postulates:

  1. ASA (Angle-Side-Angle): Two angles and the included side of one triangle are equal to two angles and the included side of another triangle.

  2. SAS (Side-Angle-Side): Two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.

  3. SSS (Side-Side-Side): Three sides of one triangle are equal to three sides of another triangle.

ASA Geometry or ASA Postulate

  • Definition: The ASA postulate states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the triangles are congruent.

  • Example of ASA:

    • Given triangles with sides denoted as AC and DF, angles as α, θ, and β, ϕ:

    • When the following conditions hold:

    • AC = DF

    • ∠A = ∠θ

    • ∠B = ∠β

    • The triangles are congruent.

  • Numerical Example:

    • Triangle with angles: 30 degrees, 10 cm, and 75 degrees.

    • The correspondent triangle has a 30-degree angle and a 10 cm side.

    • The additional angles sum to 150 degrees, giving each of the remaining angles a measure of 75 degrees.

    • Thus, triangles are congruent.

Side-Angle-Side Theorem: SAS Postulate

  • Definition: The SAS postulate states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.

  • Example of SAS:

    • Given sides AC and DF; BC and EF; and angles α and β:

    • The triangles are congruent if:

    • AC = DF

    • ∠A = ∠α

    • BC = EF

  • Numerical Example:

    • For triangles with sides of length 5 cm and angles of 53 degrees, the unknown side measures 3 cm.

    • Therefore, both triangles have corresponding side lengths that are congruent, confirming congruence.

Side-Side-Side Theorem: SSS Postulate

  • Definition: The SSS postulate states that if three pairs of sides of one triangle are equal to three pairs of sides of another triangle, then the triangles are congruent.

  • Example of SSS:

    • Given the sides AB, DE, BC, EF, and AC, DF:

    • The triangles are congruent if:

    • AB = DE

    • BC = EF

    • AC = DF

  • Illustrative Example:

    • Measuring would show correspondent sides of length equal to each other, indicating congruence under SSS.

Examples and Applications

  • Application of congruency concepts can be visually validated and calculated through geometry.

  • Example images throughout the lesson illustrate congruency through diagrammatic representations.

Comparison and Summary

  • Difference between SAS and SSS:

    • SAS involves two sides and the included angle, while SSS involves only sides.

  • Distinction of Postulates:

    • ASA connects two angles with the side in between.

    • AAS involves the angle coming after the sequence rather than between.

Conclusion

  • This lesson comprehensively covers the concept of triangle congruency and the corresponding postulates.

  • The understanding of these geometric principles aids in problem-solving and is crucial for advanced studies in geometry.

Learning Outcomes

  1. Understand the concept and definitions of congruency in triangles.

  2. Apply the ASA, SAS, and SSS postulates effectively.

  3. Differentiate between similar and congruent triangles.

  4. Analyze and solve problems involving triangle congruency.