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Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar

(Proposition 6 from Book 6 of Euclid's “Elements”)

If two triangles have one angle equal to one angle, and the sides about the equal angles proportional, then the triangles will be equiangular, and will have the angles which corresponding sides subtend equal.

• Let $ABC$ and $DEF$ be two triangles having one angle, $BAC$, equal to one angle, $EDF$ (respectively), and the sides about the equal angles proportional, (so that) as $BA$ (is) to $AC$, so $ED$ (is) to $DF$.
• I say that triangle $ABC$ is equiangular to triangle $DEF$, and will have angle $ABC$ equal to $DEF$, and (angle) $ACB$ to $DFE$.

Modern Formulation (Special Case of the "Side-Angle-Side" Theorem for the Similarity of Triangle)

Two triangles ($\bigtriangleup{ABC}$,$\bigtriangleup{DEF}$) are similar if: * they have one congruent angle ($\angle{BAC}\cong\angle{EDF}$) * and two corresponding sides are proportional: $$\frac{|\overline{AB}|}{|\overline{AC}|}=\frac{|\overline{DE}|}{|\overline{DF}|}.$$

Table of Contents

Proofs: 1

Mentioned in:

Proofs: 1 2 3 4 5

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016