Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles

Euclid's Formulation

If there are two equal rectilinear angles, and raised straight lines are stood on the apexes of them, containing equal angles respectively with the original straight lines (forming the angles), and random points are taken on the raised (straight lines), and perpendiculars are drawn from them to the planes in which the original angles are, and straight lines are joined from the points created in the planes to the (vertices of the) original angles, then they will enclose equal angles with the raised (straight lines). * Let $BAC$ and $EDF$ be two equal rectilinear angles. * And let the raised straight lines $AG$ and $DM$ have been stood on points $A$ and $D$, containing equal angles respectively with the original straight lines. (That is) $MDE$ (equal) to $GAB$, and $MDF$ (to) $GAC$. * And let the random points $G$ and $M$ have been taken on $AG$ and $DM$ (respectively). * And let the $GL$ and $MN$ have been drawn from points $G$ and $M$ perpendicular to the planes through $BAC$ and $EDF$ (respectively). * And let them have joined the planes at points $L$ and $N$ (respectively). * And let $LA$ and $ND$ have been joined. * I say that angle $GAL$ is equal to angle $MDN$.


Modern Formulation

(not yet contributed)

Proofs: 1 Corollaries: 1

Thank you to the contributors under CC BY-SA 4.0!



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",, 2016