Proposition 15
If two straight lines cut each other, they make vertical angles equal to one another.
This proposition appears a great deal in today’s geometry & trigonometry classes: Vertical angles are equal.
Let two lines AB and CD cut each other, intersecting at point E.
We will show that angle AEC is equal to angle BED, and that angle AED is equal to angle BEC.
Line AE lies on the straight line, so by Proposition 13 (When a straight line lies on a straight line, the two adjacent angles are equal to two right angles) the two angles AEC and AED are equal to two right angles.
Using the same reasoning, the two angles AED and BEDare equal to two right angles since the line DE lies on the straight line AB.
Since both pairs of angles are equal to two right angles, we know that the two angles AEC and AED are equal to the two angles AED and BED. Subtract the angle AED from each, and we find that angle AEC is equal to angle BED.
Can you prove that angle AED is equal to angle BEC? Try it first on your own, then click on the Proof Strategy tag below.
Please feel free to leave any questions/comments for me – George
1. Show that angles AED and BED are equal to two right angles.
2. Show that angles BEC and BED are equal to two right angles.
3. Conclude that the two angles AED and BED are equal to the two angles BEC and BED.
4. Conclude that angle AED is equal to angle BEC.