11.3 Vertically opposite


When 2 straight lines cross each other, 4 angles are formed. We will call these a, b, c, d as per this diagram:

Vertically opposite angles are equal to each other. In other words, angles a=c, and b=d.

Why? We know angles on a straight line add up to 180°. That means a+d=180\textdegree. We also know that d+c=180\textdegree. We can reshuffle the first formula such that d=180-a, and the second formula to d=180-c. Therefore, we know that 180-a=180-c. Simplifying, -a=-c, meaning a=c. Quod erat demonstrandum.

“Hey, what do those weird Greek letters mean?” Mandy asked.

“It’s telling other mathematicians you’ve set out what you intended to prove,” Jamie responded, “essentially a sign of relief!”

Angles such as a and c, or b and d are called vertically opposite, because they are opposite the vertex where the 2 lines meet.

Vertically opposite angles are equal to each other.


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