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2.3 – Congruence

Objectives

  • Make valid congruence statements about two triangles.
  • Discover and apply CPCTC (corresponding parts of congruent triangles are congruent).
  • Use CPCTC to determine information about congruent triangles.
  • Define the reflexive, transitive, and symmetric properties of congruence.
  • Define and apply the three congruence transformations.

 

Key Terms

  • Congruence Statement – A statement that tells which sides or angles of two triangles are congruent
  • Congruence Transformation – An action that can be performed on a geometric object without changing its size or shape.
  • After a congruence transformation, the perimeter of a triangle would be the same as it was before.
    • Rotations (turns): rotating
    • Translations (slides): translating
    • Reflections (flips): reflecting
  • Reflexive Property – The triangles are the exact same, with corresponding parts in the same order
  • Symmetric Property – The triangles have been flipped (like in a mirror), so they are backwards from one another (like butterfly wings)
  • Transitive Property – A syllogism (chain of events) involving triangles

Geo A 2.3 - Properties

  • Congruent Triangles – Triangles with all congruent sides (equal in length) and all congruent corresponding angles (equal in measure).
    • Congruent triangles have the same size and shape
    • Congruent triangles have the same angle measures and side lengths
    • The corresponding sides are congruent.
    • The corresponding angles are congruent.
  • CPCTC – Congruent Parts of Congruent Triangles are Congruent
    • If 2 triangles are congruent, then all of their parts are congruent
    • If all of the parts of two triangles are congruent, then the 2 triangles are congruent

Geo A 2.3 - Congruence

 

Notes

Congruent Triangles
  • If two triangles are congruent, then they can be moved so that they line up perfectly.
  • You can find the corresponding parts of two congruent triangles by aligning them perfectly on top of each other.
  • Congruent triangles may be flipped over or rotated (and can be lined up on top of one another).
    • To figure out if two triangles are congruent, look at the order of the letters in one triangle and match them to the order in the other triangle.

Geo A 2.3 - Congruent Triangles 01

\Delta ZXY \cong \Delta CAB

    • Notice that the letters ZXY can be traced counter-clockwise, but the triangles are flipped; so, you need to start with C on the 2nd triangle and trace clockwise to match congruence.
    • If you just look at the letters in the congruence statement, you will see that Z & C are both first, X & A are both second, and B & Y are both third.
    • Congruent Sides
      • \overline{ZX} \cong \overline{CA}
      • \overline{XY} \cong \overline{AB}
      • \overline{YZ} \cong \overline{BC}
    • Congruent Angles
      • \angle Z \cong \angle C
      • \angle X \cong \angle A
      • \angle Y \cong \angle B

  • Example (without drawings)
    • If the following triangles are congruent, what do you know about their corresponding parts?
    • \Delta GHI \cong \Delta LMN
      • Congruent Sides
        • \overline{GH} \cong \overline{LM}
        • \overline{HI} \cong \overline{MN}
        • \overline{GI} \cong \overline{LN}
      • Congruent Angles
        • \angle G \cong \angle L
        • \angle H \cong \angle M
        • \angle I \cong \angle N

  • Example with Side Lengths
    • Assume \Delta ABC \cong \Delta DEF
    • If AB = 12, BC = 15, and AC = 17, what is the length of \overline{EF}?
    • Step 1: Draw two congruent triangles
    • Step 2: Label the triangles with corresponding verticies
    • Step 3: Label the side lengths
    • Step 4: Compare the triangles to see what corresponding sides are missing their labels (lengths)
      • If AB = 12, then DE = 12
      • If BC = 15, then EF = 15
      • If AC = 17, then DF = 17
    • Step 5: Write your answer
      • \overline{EF}=15

  • Example of Property Congruence (without drawings)
    • Which property is illustrated by the following statement?
    • If \Delta HXA \cong \Delta RGI, then \Delta RGI \cong \Delta HXA
      • Answer: Symmetric Property
      • Reason: The “if” statement is the original congruence, and the “then” statement shows the triangle congruence has flipped!  Flipped congruence is Symmetric (like butterfly wings)

 

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