G6 Review: Triangle Congruence Notes

G6 Review: Triangle Congruence Notes

Triangle Congruence Overview

  • Triangle congruence allows one to deduce properties about triangles based on certain relationships.
  • Important to correspond parts of triangles as indicated by the order in congruence statements.

Key Statements Given Congruence

  • If ABC ≅ ARST, then:
    • a. AB ≅ RS
    • b. ZC ≅ LT
    • c. ZSRT ≅ BAC
    • d. TS ≅ CB
    • e. Is AC ≅ ABATSR? No

Solving for Variables in Triangle Dimensions

  • Triangle dimensions and their relationships provided:

    • For triangle AABC:
    • AB = 7
    • B = 60
    • AC = 2y
    • BC = 3x-2

  • For triangle ARST:

    • RS = 8
    • S = R
    • ST = 72
    • RT = 5
    • Z = 72
    • A = 60°
Variable Solutions
  • For AABC and ARST:
    • $y = 3.5$ (solving: 72 = 3x)
    • $x = 18$ (solving: 70 = 3x-2)
    • $W = 2.25$
    • $4x = 72 \, \Rightarrow \, x = 18$

Methods for Proving Triangle Congruence

  • There are five principal methods to prove two triangles congruent:

    1. SSS (Side-Side-Side): All corresponding sides are equal.
    • Example: Triangle sides marked would show equality:
    • AB = RS, BC = ST, AC = RT
    1. SAS (Side-Angle-Side): Two sides and the included angle are equal.
    • Example: Lengths and angle marked accordingly.
    1. ASA (Angle-Side-Angle): Two angles and the included side are equal.
    • Example: Angles A and B mark correspondingly to angles in the second triangle.

    1. AAS (Angle-Angle-Side): Two angles and a non-included side are equal.

    2. HL (Hypotenuse-Leg): For right triangles, if the hypotenuse and one leg are equal.

Non-Conclusive Methods

  • The two methods that do not work for proving triangle congruence are:
    1. AAA (Angle-Angle-Angle)
    2. SSA (Side-Side-Angle)

Practice Identifying Congruent Triangles

  • Use different methods of congruence to analyze pairs of triangles:
    • Example statements:
    • Is ACB ≅ ACD? - Yes/No (using respective methods)
    • Is ABD ≅ ABE? - Yes/No
  • Assess pairs for congruence via SSS, SAS, ASA, AAS, HL.

Completing Proofs

  • E.g. Proof Steps:

    • Given: ZNZP; MO = QO
    • Prove: ΔMON ≅ ΔQOP
    • Statements to prove include:
      1. Z NZP
      2. MO = QO
    • Use reasons like Given, Vertical Angles Theorem, and congruence definitions (AAS, SAS).

  • Another example:

  • Given:

    • AB || DE; ( C ) is the midpoint of ( AE )
    • Prove: ΔABC ≅ AEDC
    • Complete using definitions of midpoint and angle properties.

Additional Proof Techniques

  • Practice understanding of given statements and applying appropriate geometric theorems to achieve congruence.
  • Be familiar with all properties and theorems used in proofs to ensure comprehensive knowledge for proof completion.