7.3 Criteria for Congruence of Triangles

Description

Quick Overview

This section explores the specific criteria that determine when two triangles are congruent.

Standard

In this section, we recap the specific criteria for triangle congruence, including SAS, ASA, AAS, SSS, and RHS rules. Examples are provided to illustrate how these criteria apply, and exercises challenge the understanding of congruence in triangles.

Detailed

Detailed Summary

In the study of triangles, congruence is a fundamental concept that signifies two figures having the same shape and size. This section revisits the criteria for establishing the congruence of triangles, which include:

  1. Side-Angle-Side (SAS) Criterion: This states that two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.
  2. Angle-Side-Angle (ASA) Criterion: Two triangles are congruent if two angles and the included side of one triangle are equal to the two angles and the included side of the other triangle.
  3. Angle-Angle-Side (AAS) Criterion: This criterion holds true when two angles and a non-included side of one triangle are equal to the corresponding parts of another triangle.
  4. Side-Side-Side (SSS) Criterion: This implies that if all three sides of one triangle are equal to all three sides of another triangle, the triangles are congruent.
  5. Right angle-Hypotenuse-Side (RHS) Criterion: Applicable in right triangles, this declares that if the hypotenuse and one side of one right triangle are equal to the hypotenuse and one side of another right triangle, then the two triangles are congruent.

Examples in this section provide practical applications of these criteria, thereby enhancing understanding. Exercises are presented to reinforce these concepts through problem-solving and verification of congruence in various scenarios.

Key Concepts

  • SAS Criterion: Two triangles are congruent if two sides and the included angle are equal.

  • ASA Criterion: Two triangles are congruent if two angles and the included side are equal.

  • AAS Criterion: Two triangles are congruent if two angles and a non-included side are equal.

  • SSS Criterion: Two triangles are congruent if all three sides are equal.

  • RHS Criterion: In right triangles, if the hypotenuse and one side are equal, then the triangles are congruent.

Memory Aids

🎵 Rhymes Time

  • If two sides and the angle in between align, SAS will proclaim congruence just fine.

📖 Fascinating Stories

  • Once upon a time in Triangle Town, there were triangles with sides and angles. Whenever two angles and a side connected, they would celebrate their congruent friendship with AAS!

🧠 Other Memory Gems

  • For remembering the criteria: 'SAS for sides and angle in the middle, ASA for angles and side, that’s a riddle.'

🎯 Super Acronyms

You can use SSS, SAS, ASA and AAS, plus RHS for right triangles, that’s how we measure the congruence gauge.

Examples

  • Example applying SAS: If triangle ABC has sides AB = 5 cm, AC = 7 cm and angle A = 60°, it is congruent to triangle DEF with the same dimensions.

  • Example applying AAS: Triangle XYZ has angles of 50° and 60° with side XY = 10 cm. If triangle PQR has angles of 50° and 60° and side PQ = 10 cm, then they are congruent.