Key takeaway: Triangle congruence has four shortcuts, SSS, SAS, ASA and AAS, each showing that three well-chosen pairs of equal parts are enough to prove two triangles are exactly the same size and shape.
Grade 10 – GeometryG-CO.B.8Geometry and Measurement
Definition of Triangle Congruence
Triangle congruence can be proved without measuring all six parts. Four recipes with three well-chosen pairs do the job.
SSS needs every side matched. SAS needs a matched pair of sides plus the angle they enclose. ASA needs a matched pair of angles plus the side they enclose. AAS needs a matched pair of angles plus a side outside that corner. Position counts: for SAS, the angle must be wedged between the two sides, not off to one side.
Formula and notation
Each rule is a three-letter code read in order around the triangle:
\[ \text{SSS},\quad \text{SAS},\quad \text{ASA},\quad \text{AAS} \]
S marks a matched side and A a matched angle. Triangle congruence cannot be shown with the patterns SSA or AAA.
Key facts
- The matched angle is enclosed by the two sides in SAS; the matched side is enclosed by two angles in ASA.
- AAA only shows the triangles have the same shape, not the same size.
- SSA can fail because two different triangles may fit the same data.
- For right triangles there is also the HL shortcut.
Triangle Congruence example
In \(\triangle ABC\), \(AB = 9\) in., \(BC = 12\) in. and \(m\angle B = 40^{\circ}\). In \(\triangle DEF\), \(DE = 9\) in., \(EF = 12\) in. and \(m\angle E = 40^{\circ}\). Are the triangles congruent, and why?
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Match the parts: \(AB\) to \(DE\), \(BC\) to \(EF\), \(\angle B\) to \(\angle E\).
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The two equal sides meet at vertex \(B\), and \(\angle B\) is the angle between them, so it is an included angle.
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Two sides plus the included angle match the SAS rule.
Yes: \(\triangle ABC \cong \triangle DEF\) by SAS.
Common mistake
Watch out
Two triangles both have angles of \(35^{\circ}\), \(55^{\circ}\) and \(90^{\circ}\), so Ines declares them congruent by AAA.
Why it is wrong
AAA is not a rule. One triangle with a 5 cm hypotenuse and another with a 10 cm hypotenuse can share all three angles yet differ in size. Equal angles prove similarity; for congruence, at least one pair of equal sides is required.
Related terms and lessons
- Congruent Triangles
- HL Theorem (Hypotenuse-Leg)
- CPCTC
- Similar Triangles
- AA Similarity
- Two-Column Proof
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