Euclid's Elements- Book I- Proposition 15
If two straight lines cut one another, they make the vertical angles equal to one another.
In other words,
when two straight lines intersect, they form congruent opposite (vertical) angles.
1. Construct line AB.
2. Construct line CD such that it intersects line AB.
3. Label the point of intersection of both lines point E.
We want to show i. m∠AEC=m∠DEB and
ii. m∠AED=m∠BEC
i. Line AE cuts line CD, making the angles AEC and AED two right angles or equal to two right angles.
So, m∠AEC+m∠AED=180°. [I.13]
Also, line DE cuts line AB making the angles AED and DEB two
right angles or equal to two right angles.
So, m∠DEB+m∠AED=180°. [I.13]
So, if m∠AEC+m∠AED=180° and
m∠DEB+m∠AED=180°, then
m∠AEC+m∠AED=m∠DEB+m∠AED (Post 4 and C.N 1)
Hence, m∠AEC=m∠DEB by subtraction and C.N 3.
ii. Similarly, we can show 2. m∠AED=m∠BEC.
Line BE cuts line CD making angles DEB and
CEB two right angles or equal to two right angles.
Aso, m∠DEB+m∠CEB=180° [1.13].
Also, line DE cuts line AB making the angles AED and DEB two right angles or equal to two right angles
So, m∠AED+m∠DEB=180° [1.13].
So, if m∠DEB+m∠CEB=180° and m∠AED+m∠DEB=180°, then
m∠DEB+m∠CEB=m∠AED+m∠DEB (Post 4 and C.N. 1)
Hence, m∠CEB=m∠AED by subtraction and C.N 3.
Q.E.D.