Angles and Arcs
In geometry, we need to be able to prove whether two shapes are different or the same (congruent). For that, we need a formal definition of congruence for each shape we study. We've been talking about arcs and circles for a while now without a formal definition. But we'll get it out of those jeans and sneakers and into some dress shoes and a tux. How's that for formal?
If two circles have congruent radii, then they are congruent circles. If two arcs are both equal in measure and segments of congruent circles, then they are congruent arcs.
Notice that two arcs of equal measure that are part of the same circle are congruent arcs, since any circle is congruent to itself.
Let's circle back (pun intended) to the track example. Is the arc you travel as you run in the inner lane of the track congruent to the arc your friend travels as he runs in the outer lane? No, because the two arcs are not segments of congruent circles. They have different radii.
However, the original question asked whether you and your friend run the same distance. That question is about arc length, not arc congruence.
Congruent arcs have equal length (you can prove this yourself). Does that mean all arcs of equal length are congruent? Nope. (You can prove this yourself too.) That's like saying, "All cars can travel at 65 miles an hour, so everything that travels 65 miles an hour is a car." That's untrue, not to mention insulting to a good number of cheetahs.
We can relate central angles to arcs using the Angle-Arc Theorem: In congruent circles, two central angles are congruent if and only if their intercepted arcs are congruent. This is a biconditional statement, meaning that it goes both ways.
The first way: If two arcs are congruent, then the two central angles that intercept them are congruent. The second way: If two central angles are congruent, then the arcs they intercept are congruent.
To prove a biconditional statement, we have to prove the statement in both directions. In other words, we have to prove two statements. Let's start with the first one.
We're given that ⊙O is congruent to ⊙O' and arc AB is congruent to arc A'B'. To prove that ∠AOB is congruent to ∠A'O'B', we can say that by the definition of congruence of arc, mAB = mA'B'. By definition of arc measure, m∠AOB = m∠A'O'B'. By definition of congruence of angle, ∠AOB is congruent to ∠A'O'B'. Unfortunately, we can't just recite this backwards to satisfy the other part of the biconditional.
If ∠AOB is congruent to ∠A'O'B', that tells us m∠AOB = m∠A'O'B'. By definition of arc measure, mAB = mA'B'. We're also given that ⊙O is congruent to ⊙O'. Since arcs AB and A'B' have the equal measure and are segments of congruent circles, we can say by definition of congruent arcs that arcs AB and A'B' are congruent.
That "if-and-only-if" part makes a statement much stronger because it's fortified from both ends. It's like a multivitamin for mathematical statements. Only without that horrible lodged-in-your-throat feeling.
One more thing about arcs before we move on. We can add them, just like we can add numbers. It seems silly to add shapes, doesn't it? What does that even mean?
For those deep, deep questions such as "what does arc addition mean," we need something more than a simple definition. Enter postulates. More specifically, the Arc Addition Postulate.
Given two arcs in the same circle AB and BC with exactly one point in common (the endpoint B), we say: arc AB + arc BC = arc ABC. Of course, this also means mAB + mBC = mABC.
Arc addition will come in handy later. Trust us.
Also, don't get overambitious with postulates. They're helpful and all, but they can't answer every question for us. For instance, it's usually a bad strategy to write, "I postulate that I will get full credit on this exam." You'll probably end up with zero credit and a massive grounding.