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.

$\Delta ZXY \cong \Delta CAB$

• 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
• $\overline{EF}=15$
• $If \Delta HXA \cong \Delta RGI, then \Delta RGI \cong \Delta HXA$