Angles at a point sum to 360
Splitting a full turn in four, and why the pieces always add back up to 360.
Last lesson, one ray split a straight angle in two. This lesson asks what happens when a whole turn around a point is split up instead.
Here is a straight line through A and C, with O sitting exactly between them, just like B was last lesson. Two rays run out from O: one to D, above the line, and one to E, below it, together splitting the full turn around O into four angles. Each of D and E stays within its own half of the line as you drag.
A straight line through A and C, with O exactly between them, and two rays out to D and E splitting the full turn around O into four angles. D and E can be dragged. This needs JavaScript switched on.
Dragging D onto the line itself makes one of its two angles read 180° and the other 0°, and the running total is still 360.
The vocabulary for four angles at a point
O is the vertex all four angles share. Angle AOD and angle DOC sit above the line; angle COE and angle EOA sit below it. Any group of angles at a single point that together account for a full turn are said to be angles at a point, and a full turn itself is 360 degrees.
Why the four angles always total 360
Angle AOD, angle DOC, angle COE and angle EOA are four entirely separate measurements. What makes them add to 360 every time is the same single fact from two lessons ago, used twice: OA and OC point in exactly opposite directions, because O is the midpoint of a straight line through A and C.
That opposite-directions fact is exactly what made angle AOD and angle DOC sum to 180 last lesson, whatever direction OD pointed in. It applies again, completely separately, to OE: angle COE and angle EOA sum to 180 too, whatever direction OE points in. Add the two straight angles together and the full turn around O is
however far apart or close together D and E are dragged.
What this sets up
This is the pattern every angle-sum fact in the course will follow from here: take a known total, split it, and reapply what already summed correctly on each piece separately.
The next lesson asks about the two angles directly across the line from each other, angle AOD and angle COE’s opposite pair, rather than about a running total. It is the first lesson in the course that needs two separate lines of reasoning to reach an answer that is not obvious in advance.
Angle AOD measures 70°. What is angle DOC, given that A, O and C lie on a straight line?
Angle COE measures 50°. What is angle EOA?
Angle AOD measures 70°, angle DOC measures 110°, and angle COE measures 50°. What is angle EOA, using only the fact that all four angles at O sum to 360?
which matches the 130° found directly above, because both are the same fact used two different ways.
This lesson counts as done once you get its exercise right. Nothing to tick off by hand.