Proposition VI.7 — If two triangles have one angle equal to one angle, the side…

Proof

Construct, at the vertex of the angle whose sides are proportional, an angle equal to the corresponding angle in the second triangle (I.23). The resulting auxiliary triangle agrees with the first in two angles (and hence all three, by I.32) and shares a side with the second; the constraint on the remaining angle being acute or obtuse ensures the construction is non-ambiguous (essentially eliminates the SSA failure case).

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Depends on (4)

I.23Proposition I.23On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.
I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles,…
VI.4Proposition VI.4In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides which…
VI.6Proposition VI.6If two triangles have one angle equal to one angle and the sides about the equal angles proportional, the triangles…

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