---
book: 1
number: 15
id: "I.15"
kind: "theorem"
uses: ["[[book-1/proposition-13]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.15"
license: "CC-BY-SA-4.0"
---

# I.15

If two straight lines cut one another, they make the vertical angles equal to one another.

![I.15](figures/I-15.svg)

## Proof

For let the straight lines *AB*, *CD* cut one another at the point *E*;

I say that the angle *AEC* is equal to the angle *DEB*, and the angle *CEB* to the angle *AED*.

For, since the straight line *AE* stands on the straight line *CD*, making the angles *CEA*, *AED*, the angles *CEA*, *AED* are equal to two right angles [[book-1/proposition-13|I. 13]]

Again, since the straight line *DE* stands on the straight line *AB*, making the angles *AED*, *DEB*, the angles *AED*, *DEB* are equal to two right angles. [[book-1/proposition-13|I. 13]]

But the angles *CEA*, *AED* were also proved equal to two right angles; therefore the angles *CEA*, *AED* are equal to the angles *AED* *DEB*. [[book-1/postulates#Postulate 4|Post. 4]] and [[book-1/common-notions#Common Notion 1|C. N. 1]] Let the angle *AED* be subtracted from each; therefore the remaining angle *CEA* is equal to the remaining angle *BED*. [[book-1/common-notions#Common Notion 3|C. N. 3]]

Similarly it can be proved that the angles *CEB*, *DEA* are also equal.

Therefore etc. Q. E. D.

[Porism. From this it is manifest that, if two straight lines cut one another, they will make the angles at the point of section equal to four right angles.