Math 2.0

Geometry Theorem

Angle-angle similarity

Builds on Similar trianglesAngle at a pointAngle sum of a triangle

Statement

Let ABCABC and A′B′C′A'B'C' be triangles in the Euclidean space ℝn\mathbb{R}^n, and let ∠\angle denote the angle at a point. If ∠BAC=∠B′A′C′\angle BAC = \angle B'A'C' and ∠CBA=∠C′B′A′\angle CBA = \angle C'B'A', then ABCABC and A′B′C′A'B'C' are similar.

Remarks

By the angle sum of a triangle, the third angles are then equal too. So the criterion is also called angle-angle-angle similarity. Another criterion for similarity is side-angle-side: two equal ratios of sides, and equal angles between those sides. In hyperbolic geometry, triangles with equal angles are congruent, so the criterion has a different meaning there.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph
Angle at a point, concept, no evidence levelAngle at a pointconceptSimilar triangles, concept, no evidence levelSimilar trianglesconceptTriangle, concept, no evidence levelTriangleconceptAngle sum of a triangle, statement, no evidence levelAngle sum of a triang…statementAngle-angle similarity, statement, no evidence levelAngle-angle similaritystatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Angle at a point concept
  2. Similar triangles concept
  3. Triangle concept
  4. Angle sum of a triangle statement
  5. Angle-angle similarity statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Similarity_(geometry), Similar triangles
About this page
Tag0NZVJYZVTZ
URIm2://0NZVJYZVTZ
Aliasesaa-similarity, aaa-similarity
Kindstatement, theorem
WritingW0 stub
MSC51M04
StatusThis copy of the site has no build data. Run m2 status 0NZVJYZVTZ.
UsesSimilar triangles, Angle at a point, Angle sum of a triangle

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