Intro to Triangle Congruence
Two triangles are congruent when they are exactly the same shape and size — every pair of corresponding sides and angles match. The key is identifying which parts correspond.
Introduction
In Chapter 4 you learned that rigid motions (translations, reflections, rotations) preserve size and shape. Two figures are congruent if one can be mapped onto the other by a sequence of rigid motions. For triangles, this means 6 parts must match: 3 sides and 3 angles.
Past Knowledge
Rigid motions (Ch 4). Triangle properties (5.1). Angle Sum Theorem.
Today's Goal
Understand congruence, identify corresponding parts, and write congruence statements.
Future Success
SSS, SAS, ASA, AAS congruence shortcuts (5.2.2–5.2.5), and CPCTC proofs.
Key Concepts
Definition of Congruent Triangles
Two triangles are congruent () if and only if all six pairs of corresponding parts are congruent:
Corresponding Sides
Corresponding Angles
The Congruence Statement
Order matters! When we write , the vertex order tells us which parts correspond:
Third Angles Theorem
If two angles of one triangle are congruent to two angles of another, then the third angles are also congruent. (This follows from the Triangle Angle Sum Theorem — both thirds equal .)
Worked Examples
Writing a Congruence Statement
has and has . Write a valid congruence statement.
Match the sides by length:
(or vice-versa)
Wait — let's be systematic. , , .
So:
Finding Missing Parts
Given , , , and . Find and .
From the congruence statement:
Third Angles Theorem
In and , , , , . What can you conclude about and ? Does this prove the triangles are congruent?
and
By the Third Angles Theorem:
But this does NOT prove congruence! All angles match, but the triangles could be different sizes (similar, not congruent). We need at least one pair of corresponding sides to match.
, but congruence requires side information too.
Common Pitfalls
Writing Vertices in the Wrong Order
is not the same as . The order of vertices determines which parts correspond. Always check that matching vertices are in matching positions.
Thinking AAA Proves Congruence
Three pairs of equal angles only proves the triangles are similar (same shape). They could be different sizes. To prove congruence, you must show at least one pair of corresponding sides is equal.
Real-Life Applications
Manufacturing — Quality Control
When a factory produces triangular components (brackets, gusset plates, etc.), every piece must be congruent to a master template. Inspectors verify corresponding measurements to ensure interchangeability.
Carpentry — Duplicate Cuts
Carpenters use a jig (template) to cut multiple identical triangular pieces. The jig guarantees that all corresponding sides and angles match — literal triangle congruence ensuring every rafter in a roof frame is identical.
Practice Quiz
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