Duplicated Triangle Forms a Kite or a Parallelogram

Theorem

Let a triangle be copied either by rotation or reflection.

Then if you mung them together by corresponding sides what you get is either a parallelogram or a kite.

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Proof

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Let $\triangle ABC$ be any triangle. We can form a congruent triangle by reflection or translation.

Case $1$: Reflection

Reflect $\triangle ABC$ across one of its sides.

Without loss of generality draw the two triangles with the longest side shared.

The resulting $\Box ABCD$ has two pairs of equal sides adjacent (it forms a kite).

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Case $2$: Translation

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Translate a second congruent copy of $\triangle ABC$, rotate it.

Without loss of generality draw the two triangles together with the longest side shared.

The resulting $\Box ABCD$ has two pairs of opposite sides equal.

By Quadrilateral is Parallelogram iff Both Pairs of Opposite Sides are Equal or Parallel:

$\Box ABCD$ is a parallelogram

By definition of parallelogram:

Both pairs of opposite angles in $\Box ABCD$ are equal

Special case: Right Triangle

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Consider the special case where $\triangle ABC$ is a right triangle with $\angle ABC$ and $\angle ABD$ both right angles.

Then $\Box ABCD$ has opposite sides parallel and both pairs of opposite angles equal.

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Since one pair of angles is a pair of right angles:

the other pair consists of two right angles

Therefore by the definition of rectangle:

$\Box ABCD$ is a rectangle.

$\blacksquare$