Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms
(Proposition 34 from Book 1 of Euclid's “Elements”)
In parallelogrammic figures the opposite sides and angles are equal to one another, and a diagonal cuts them in half.
- Let $ACDB$ be a parallelogrammic figure, and $BC$ its diagonal.
- I say that for parallelogram $ACDB$, the opposite sides and angles are equal to one another, and the diagonal $BC$ cuts it in half.
Modern Formulation
The opposite sides and the opposite angles of a parallelogram are equal to one another and either diagonal bisects the parallelogram. In particular, the area of the parallelogram \(\boxdot{ABDC}\) is double the area of \(\triangle{ACB}\), (respectively \(\triangle{BCD}\)).
Table of Contents
Proofs: 1 Corollaries: 1 2 3 4 5 6
- Definition: Parallelogram - Defining Property III
Mentioned in:
Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Thank you to the contributors under CC BY-SA 4.0! 
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- @Calahan
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References
Adapted from CC BY-SA 3.0 Sources:
- Callahan, Daniel: "Euclid’s 'Elements' Redux" 2014
Adapted from (Public Domain)
- Casey, John: "The First Six Books of the Elements of Euclid"
Adapted from (subject to copyright, with kind permission)
- Fitzpatrick, Richard: Euclid's "Elements of Geometry"
