7.2 Congruence of Triangles

Description

Quick Overview

This section elaborates on the concept of congruence in triangles, discussing how triangles are congruent when their corresponding sides and angles are equal.

Standard

This section introduces the definition of congruence, outlines several examples of congruent figures, and explains the principles behind the congruence of triangles. It covers the criteria for triangle congruence, such as the SAS, ASA, and SSS rules, emphasizing the significance of corresponding parts in congruent triangles.

Detailed

Congruence of Triangles

In this section, we explore the concept of congruence, primarily focusing on triangles. Congruent figures are those that have identical shapes and sizes. The section highlights that two triangles are congruent if their corresponding sides and angles are equal. Various criteria for triangle congruence are introduced, including:

  1. SAS (Side-Angle-Side) Theorem: Two triangles are congruent if two sides and the angle between them are equal.
  2. ASA (Angle-Side-Angle) Theorem: Two triangles are congruent if two angles and the side between them are equal.
  3. SSS (Side-Side-Side) Theorem: Two triangles are congruent if all three sides of one triangle are equal to all three sides of another triangle.
  4. AAS (Angle-Angle-Side) Theorem: Two triangles are congruent if two angles and a non-included side are equal.
  5. RHS (Right angle-Hypotenuse-Side) Theorem: If in two right triangles, the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.

The concept of congruence is essential in applications such as manufacturing, where identical pieces need to fit together perfectly. Additionally, the section emphasizes the importance of consistent notation when denoting congruent triangles, which includes proper correspondence of their vertices. The section concludes with exercises that reinforce the understanding of triangle congruence through practical examples.

Key Concepts

  • Congruent Figures: Figures that have identical shapes and sizes.

  • Criteria for Congruence: Rules that help determine whether two triangles are congruent, including SAS, ASA, AAS, SSS, and RHS.

Memory Aids

🎵 Rhymes Time

  • If sides and an angle are the same, then congruence is the name of the game.

📖 Fascinating Stories

  • Imagine two friends, Amy and Sally, who both have perfectly identical bags. They use them for their identical laptops. Just like their bags, congruent triangles are identical in size and shape.

🧠 Other Memory Gems

  • For triangle congruence remember: SAS, ASA, AAS, SSS. Just think 'Silly Animals Sing Sweetly.'

🎯 Super Acronyms

Use the acronym 'CATS' to remember congruence rules

  • Corresponding parts
  • All sides
  • Triangles
  • Same angles.

Examples

  • Two triangles with both pairs of corresponding sides and angles equal are congruent by the criteria of SAS.

  • If two triangles share a side and have two equal angles adjacent to that side, they are congruent by the ASA criterion.

Glossary of Terms

  • Term: Congruence

    Definition:

    The property of figures having the same shape and size.

  • Term: SAS Criterion

    Definition:

    A rule stating that two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.

  • Term: ASA Criterion

    Definition:

    A rule that states that two triangles are congruent if two angles and the side between them are equal in both triangles.

  • Term: SSS Criterion

    Definition:

    A statement that if three sides of one triangle are equal to three sides of another triangle, the two triangles are congruent.

  • Term: AAS Criterion

    Definition:

    A theorem stating that if two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another triangle, the triangles are congruent.

  • Term: RHS Criterion

    Definition:

    A rule applying specifically to right triangles; states that if the hypotenuse and one side are equal, the triangles are congruent.

  • Term: CPCT

    Definition:

    Corresponding Parts of Congruent Triangles, meaning that the parts of congruent triangles are equal.