Angle On A Straight Line

Theorem: Angle on each side of a straight line is always equal to $180^o$.

Prerequisites:
Degree Scale (definition)

Proof:

Let there be a circle '$c$'. Let the line '$l$' pass through the center of the circle $c$ and forms the angles '$\alpha$' and '$\beta$' in the two parts of the circle. Now due to the symmetry of the circle, the line $l$ divides the circle into two equal halves.

$\therefore \qquad\quad\angle\alpha = \angle\beta\qquad\qquad\qquad\qquad\qquad\qquad\cdots (1)$

Also, by the definition of degree-scale of the angle, central angle of the circle $= 360^o$.

$\therefore\qquad\quad\angle\alpha + \angle\beta = 360^o$
$\Rightarrow\qquad\quad\!\! 2\angle\alpha = 360^o\qquad\qquad\qquad\qquad\qquad\;\;\;\text{(from $(1)$)}$
$\Rightarrow\qquad\quad\!\!\angle\alpha = \angle\beta = 180^o$

Hence the result.


Recommended:
Angle Sum Property of triangle
Angle Sum Property of polygon
Vertical angle theorem