Triangle Congruence (SSS, SAS, ASA, AAS)
In short: Triangle congruence shortcuts, SSS, SAS, ASA and AAS, show that two triangles are identical by matching just three well-chosen parts instead of all six.
GeometryGeometryG-CO.B.8
What is Triangle Congruence?
The triangle congruence theorems save work: rather than checking all six parts of two triangles, you verify three, and four patterns of three are enough.
SSS needs three equal side pairs. SAS needs two sides with the angle trapped between them. ASA needs two angles with the side between them. AAS needs two angles and a side outside that gap. The letters follow the order of the parts around the triangle. Three equal angles prove only similarity, and side-side-angle is not a valid pattern.
How to spot it
- A figure carries tick marks on sides and arcs on angles and asks whether the two triangles match; triangle congruence is the tool.
- A proof lists a few equal parts and demands a reason before corresponding parts may be used.
- A shared side or a pair of vertical angles hides one more equal pair that you may add.
- The question names one of the four patterns, or asks which pattern the given parts allow.
The method
- List every pair of equal parts the problem or figure gives, adding shared sides and vertical angles.
- Mark the pairs in order around each triangle and note whether an angle sits between two chosen sides.
- Match the pattern to SSS, SAS, ASA or AAS; avoid SSA and AAA.
- State the conclusion with vertices in matching order, such as \(\triangle ABC \cong \triangle DEF\).
Triangle Congruence: solved example
Problem
In \(\triangle PQR\) and \(\triangle STU\), \(\angle P = \angle S = 58^{\circ}\), \(\angle Q = \angle T = 71^{\circ}\) and \(QR = TU = 14\). Which shortcut proves the triangles congruent?
Solution
Two angle pairs are given, and side \(QR\) touches \(Q\) but is not between \(P\) and \(Q\); the side between them would be \(PQ\). A non-included side with two angles is AAS.
Check with the third angle: \(180 – 58 – 71 = 51^{\circ}\) in both, so ASA could be reached too. The accepted reason is AAS, giving \(\triangle PQR \cong \triangle STU\).
Try it
Question 1
Two triangles have side lengths 6, 8 and 11. Which shortcut proves them congruent?
Show the answer
All three pairs of sides are equal, so SSS applies.
Question 2
Two triangles have two pairs of equal sides and one pair of equal angles that is not between those sides. Is that enough?
Show the answer
No. This SSA pattern can produce two different triangles, so it does not prove congruence.
Triangle Congruence vs. Similar Triangles
Similarity tests such as AA need only matching angles, since only shape must agree. A test for sameness must also fix size, so at least one pair of sides has to be equal. This explains why AAA never works here: a tiny and a huge equilateral triangle share all three angles.
Keep going
Lessons
Worksheets and quizzes
- Congruent Triangles and Rigid Motions – Worksheet
- Congruent Triangles and Rigid Motions – Quiz
- Theorems About Triangles – Worksheet
- Theorems About Triangles – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry