By definition of arc measure, m AB = m A'B'.
If ∠ AOB is congruent to ∠ A'O'B', that tells us m∠ AOB = m∠ A'O'B'. With biconditional statements, we can't always just reverse the argument to get the reverse implication, but in this case we can. By definition of congruence of angle, ∠ AOB is congruent to ∠ A'O'B'. By definition of arc measure, m∠ AOB = m∠ A'O'B'.
To prove that ∠ AOB is congruent to ∠ A'O'B', we can say that by the definition of congruence of arc, m AB = m A'B'. We're given that ⊙ O is congruent to ⊙ O' and arc AB is congruent to arc A'B'. In other words, we have to prove two statements. To prove a biconditional statement, we have to prove the statement in both directions. The second way: If two central angles are congruent, then the arcs they intercept are congruent. The first way: If two arcs are congruent, then the two central angles that intercept them are congruent. This is a biconditional statement, meaning that it goes both ways. 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. (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. Does that mean all arcs of equal length are congruent? Nope. That question is about arc length, not arc congruence.Ĭongruent arcs have equal length (you can prove this yourself). However, the original question asked whether you and your friend run the same distance. 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. Let's circle back (pun intended) to the track example. Notice that two arcs of equal measure that are part of the same circle are congruent arcs, since any circle is congruent to itself. If two arcs are both equal in measure and they're segments of congruent circles, then they're congruent arcs. If two circles have congruent radii, then they're congruent circles. But we'll get it out of those jeans and sneakers and into some dress shoes and a tux. We've been talking about arcs and circles for a while now without a formal definition. For that, we need a formal definition of congruence for each shape we study. In geometry, we need to be able to prove whether two shapes are different or the same (congruent).
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