---
book: 1
number: 14
id: "I.14"
kind: "theorem"
uses: ["[[book-1/proposition-13]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.14"
license: "CC-BY-SA-4.0"
---

# I.14

If with any straight line, and at a point on it, two straight lines not lying on the same side make the adjacent angles equal to two right angles, the two straight lines will be in a straight line with one another.

![I.14](figures/I-14.svg)

## Proof

For with any straight line *AB*, and at the point *B* on it, let the two straight lines *BC*, *BD* not lying on the same side make the adjacent angles *ABC*, *ABD* equal to two right angles;

I say that *BD* is in a straight line with *CB*.

For, if *BD* is not in a straight line with *BC*, let *BE* be in a straight line with *CB*.

Then, since the straight line *AB* stands on the straight line *CBE*, the angles *ABC*, *ABE* are equal to two right angles. [[book-1/proposition-13|I. 13]] But the angles *ABC*, *ABD* are also equal to two right angles; therefore the angles *CBA*, *ABE* are equal to the angles *CBA*, *ABD*. [[book-1/postulates#Postulate 4|Post. 4]] and [[book-1/common-notions#Common Notion 1|C.N. 1]]

Let the angle *CBA* be subtracted from each; therefore the remaining angle *ABE* is equal to the remaining angle *ABD*, [[book-1/common-notions#Common Notion 3|C.N. 3]] the less to the greater: which is impossible. Therefore *BE* is not in a straight line with *CB*.

Similarly we can prove that neither is any other straight line except *BD*. Therefore *CB* is in a straight line with *BD*.

Therefore etc.

Q. E. D.