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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
 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
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.
 Notice that the letters ZXY can be traced counterclockwise, 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
 Congruent Angles
 Example (without drawings)
 If the following triangles are congruent, what do you know about their corresponding parts?

 Congruent Sides
 Congruent Angles
 Example with Side Lengths
 Assume
 If AB = 12, BC = 15, and AC = 17, what is the length of ?
 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
 Example of Property Congruence (without drawings)
 Which property is illustrated by the following statement?

 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)

Important!
Practice (Apex 2.3)
 Practice: Pgs 8, 9, 14, 17, 18
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