---
book: 1
number: 13
id: "I.13"
kind: "theorem"
uses: ["[[book-1/proposition-11]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.13"
license: "CC-BY-SA-4.0"
---

# I.13

If a straight line set up on a straight line make angles, it will make either two right angles or angles equal to two right angles.

![I.13](figures/I-13.svg)

## Proof

For let any straight line *AB* set up on the straight line *CD* make the angles *CBA*, *ABD*;

I say that the angles *CBA*, *ABD* are either two right angles or equal to two right angles.

Now, if the angle *CBA* is equal to the angle *ABD*, they are two right angles. [[book-1/definitions#Definition 10|Def. 10]]

But, if not, let *BE* be drawn from the point *B* at right angles to *CD*; [[book-1/proposition-11|I. 11]] therefore the angles *CBE*, *EBD* are two right angles.

Then, since the angle *CBE* is equal to the two angles *CBA*, *ABE*, let the angle *EBD* be added to each; therefore the angles *CBE*, *EBD* are equal to the three angles *CBA*, *ABE*, *EBD*. [*C. N*. 2]

Again, since the angle *DBA* is equal to the two angles *DBE*, *EBA*, let the angle *ABC* be added to each; therefore the angles *DBA*. *ABC* are equal to the three angles *DBE*, *EBA*, *ABC*. [*C. N*. 2]

But the angles *CBE*, *EBD* were also proved equal to the same three angles; and things which are equal to the same thing are also equal to one another; [*C. N*. 1] therefore the angles *CBE*, *EBD* are also equal to the angles *DBA*, *ABC*. But the angles *CBE*, *EBD* are two right angles; therefore the angles *DBA*, *ABC* are also equal to two right angles.

Therefore etc.

Q. E. D.