7 TRIANGLES

Description

Quick Overview

This section introduces triangles, focusing on their properties, congruence, and the criteria for determining congruency.

Standard

The section elaborates on the fundamental aspects of triangles, including the definition of triangles, congruence, and various criteria like SAS, ASA, AAS, SSS, and RHS for triangle congruence. It also covers important properties such as the relationships between angles and sides in isosceles triangles and various exercises to solidify comprehension.

Detailed

TRIANGLES

Introduction

This section of the chapter provides a detailed exploration of triangles, highlighting their crucial properties and congruence principles. A triangle,, identified as a closed figure formed by three intersecting lines, possesses three sides, three angles, and three vertices. The section also reflects on prior studies about properties of triangles and introduces new concepts pertaining to congruence.

Congruence of Triangles

Congruent figures share equal shape and size, applicable in everyday life; for example, congruent objects found in ice trays. Triangles are congruent if their corresponding sides and angles are equal. The relationship is symbolically expressed as Ξ”PQR β‰… Ξ”ABC, where corresponding vertices match.

Criteria for Congruence

The section discusses four primary criteria for triangle congruence:
1. SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding parts of another triangle.
2. ASA (Angle-Side-Angle): Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding parts of another triangle.
3. AAS (Angle-Angle-Side): Two triangles are congruent if two angles and one non-included side are equal.
4. SSS (Side-Side-Side): Two triangles are congruent if all three sides of one triangle are equal to the corresponding sides of another triangle.
5. RHS (Right angle-Hypotenuse-Side): Applies specifically to right triangles where the hypotenuse and a side of one triangle are equal to the hypotenuse and a side of another triangle.

Additional Properties and Applications

The section covers important properties associated with isosceles triangles, particularly focusing on how angles opposite equal sides are also equal, supported by theorems and examples. Numerous exercises and applications reinforce these concepts, allowing for practical understanding of triangle properties.

Overall, the section serves as a comprehensive foundation for understanding the structure and congruence of triangles, crucial elements in geometry.

Key Concepts

  • Congruence: Two triangles are congruent if their corresponding sides and angles are identical.

  • Congruence Criteria: There are several methods to determine triangle congruence, including SAS, ASA, AAS, SSS, and RHS.

  • Isosceles Triangle: A triangle with at least two sides equal, resulting in equal opposite angles.

Memory Aids

🎡 Rhymes Time

  • Triangles have three sides true, angles too, and now you know their congruence too!

πŸ“– Fascinating Stories

  • Imagine a land of equals where trees (sides) and hills (angles) stand the same, they share their shapes and play the congruence game!

🧠 Other Memory Gems

  • To remember the congruence rules, think: 'Some Animals Are Still Running' - SAS, ASA, AAS, SSS, RHS.

🎯 Super Acronyms

CPCT

  • Corresponding Parts of Congruent Triangles are equal!

Examples

  • Example: If triangle ABC has sides of lengths 3 cm, 4 cm, and 5 cm, and triangle DEF has sides of 3 cm, 4 cm, and 5 cm as well, then Ξ”ABC β‰… Ξ”DEF by the SSS criterion.

  • Example: In an isosceles triangle where sides AB = AC, it follows that the angles opposite these sides (∠B and ∠C) are equal.

Glossary of Terms

  • Term: Triangle

    Definition:

    A closed figure formed by three intersecting lines.

  • Term: Congruence

    Definition:

    A property indicating that two figures have the same shape and size.

  • Term: Vertices

    Definition:

    The corner points of a polygon, including triangles.

  • Term: SAS (SideAngleSide)

    Definition:

    A criterion for triangle congruence where two sides and the included angle are equal.

  • Term: ASA (AngleSideAngle)

    Definition:

    A criterion for triangle congruence where two angles and the included side are equal.

  • Term: AAS (AngleAngleSide)

    Definition:

    A criterion for triangle congruence where two angles and one non-included side are equal.

  • Term: SSS (SideSideSide)

    Definition:

    A criterion for triangle congruence where all three sides are equal.

  • Term: RHS (Right angleHypotenuseSide)

    Definition:

    A criterion specifically for right triangles.

  • Term: Isosceles Triangle

    Definition:

    A triangle with at least two equal sides.