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# projection of right angle

Theorem. The projection of a right angle in $\mathbb{R}^{3}$ onto a plane is a right angle if and only if at least one of its sides is parallel to the plane.

*Proof.* Consider the projection of an angle $\alpha$ with vertex $P$ onto the plane $\pi$. Let $P^{{\prime}}$ be the projection of $P$ onto $\pi$. If neither of the sides of $\alpha$ is parallel to $\pi$, then the lines of the sides intersect the plane in two distinct points $A$ and $B$. In order to that the angle of view of the segment $AB$ seen from the point $P$ would be a right angle, $P$ must be on a sphere with diameter $AB$ centered at a point $O$. In order to that the projection angle $AP^{{\prime}}B$ would be a right angle, the point $P^{{\prime}}$ must be on a circle of the plane $\pi$ having $AB$ as diameter. But $OP^{{\prime}}$ is as the projection of the segment $OP$ shorter than $OP$. It follows that the angle $AP^{{\prime}}B$ is obtuse and hence cannot be right.

On the other hand, it’s not hard to see that the projection of a right angle is a right angle always when one or both of its sides are parallel to the projection plane.

# References

- 1 E. J. Nyström: Korkeamman geometrian alkeet sovellutuksineen. Kustannusosakeyhtiö Otava, Helsinki (1948).

## Mathematics Subject Classification

51N99*no label found*51N20

*no label found*

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