Euclid Element Book 1 proposition 7

1. Proposition - 7
Raman Choubay

2. Proposition 7• On the same straight-line, two other straight-lines
equal, respectively, to two given straight-lines which
meet cannot be constructed meeting at a different
point on the same side of the straight-line, but having
the same ends as the given straight-lines.

3. Proof
Join CD.
If possible, given two straight lines AC and CB constructed on the straight
line AB and meeting at the point C,
Let two other straight lines AD and DB be
constructed on the same straight
line AB, on the same side of it, meeting in
another point D and equal to the former
two respectively, namely each equal to that
from the same end, so
that AC equals AD which has the same
end A, and CB equals DB which has the
same end B.

4. Proof
Statements Reason
AC = AD Given
Now again DB = BC 𝐺𝑖𝑣𝑒𝑛
∠ BDC = ∠ 𝐵𝐶𝐷 𝐴𝑛𝑔𝑙𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑑 𝑏𝑦 𝑒𝑞𝑢𝑎𝑙 𝑠𝑖𝑑𝑒𝑠
In ∆ 𝐴𝐶𝐷 𝑎𝑛𝑑 ∆ 𝐵𝐶𝐷
But it was also proved much greater than it, which
is impossible.
∠ ACD = ∠ 𝐴𝐷𝐶 𝐴𝑛𝑔𝑙𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑒𝑑 𝑏𝑦 𝑒𝑞𝑢𝑎𝑙 𝑠𝑖𝑑𝑒
∠ ADC > ∠ 𝐷𝐶𝐵 ∠ 𝐷𝐶𝐵 is part of whole ∠ 𝐴𝐶𝐷
∠ BDC is much > ∠ 𝐷𝐶𝐵 ∠ ADC is the part of∠ BDC