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6.1: Similar triangles

( \newcommand{\kernel}{\mathrm{null}\,}\)

Two triangles ABC and ABC are called similar (briefly ABCABC) if (1) their sides are proportional; that is,

AB=kAB,BC=kBC and CA=kCA

for some k>0, and (2) the corresponding angles are equal up to sign:

ABC=±ABCBCA=±BCACAB=±CAB

Remarks

  • According to Theorem 3.3.1, in the above three equalities, the signs can be assumed to be the same.
  • If ABCABC with k=1 in 6.1.1, then ABCABC.
  • Note that "" is an equivalence relation. That is,
    (i) ABCABC for any ABC.
    (ii) if ABCABC, then
    ABCABC
    (iii) If ABCABC and ABCABC, then
    ABCABC

Using the new notation "", we can reformulate Axiom V:

Theorem 6.1.1 Reformulation of Axiom V.

If for the two triangles ABC, ABC, and k>0 we have B[AB), C[AC), AB=kAB and AC=kAC, then ABCABC.

In other words, the Axiom V provides a condition which guarantees that two triangles are similar. Let us formulate three more such similarity conditions.

Theorem 6.1.2 Similarity conditions

Two triangles ABC and ABC are similar if one of the following conditions holds:

(SAS) For some constant k>0 we have

AB=kAB,AC=kAC

and BAC=±BAC.

(AA) The triangle ABC is nondegenerate and

ABC=±ABC,BAC=±BAC.

(SSS) For some constant k>0 we have

AB=kAB,AC=kAC,CB=kCB.

Each of these conditions is proved by applying Axiom V with the SAS, ASA, and SSS congruence conditions respectively (see Axiom IV and the Theorem 4.2.1, Theorem 4.4.1).

Proof

Set k=ABAB. Choose points B[AB) and C[AC), so that AB=kAB and AC=kAC. By Axiom V, ABCABC.

Applying the SAS, ASA or SSS congruence condition, depending on the case, we get that ABCABC. Hence the result.

A bijection XX from a plane to itself is called angle preserving transformation if

ABC=ABC

for any triangle ABC and its image ABC.

Exercise 6.1.1

Show that any angle-preserving transformation of the plane multiplies all the distance by a fixed constant.

Hint

By the AA similarity condition, the transformation multiplies the sides of any nondegenerate triangle by some number which may depend on the triangle.

Note that for any two nondegenerate triangles that share one side this number is the same. Applying this observation to a chain of triangles leads to a solution.


This page titled 6.1: Similar triangles is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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