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Explain why two minor arcs of the same circle or of congruent circles are congruent if and only if their central angles are congruent?

Danneh Corn
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    You should show an attempt when you ask a question. What have you tried so far? – Kevin Long Dec 15 '16 at 18:33
  • Well, i don't understand this lesson at all... – Danneh Corn Dec 15 '16 at 18:35
  • To start, assume that two minor arcs of the same or congruent circles are congruent. What does it mean for them to be congruent? What does it mean about their central angles? – Kevin Long Dec 15 '16 at 18:37
  • To be congruent is that they equal each other..? Would the central angles be congruent too? I'm sorry, i'm horrible at learning this stuff – Danneh Corn Dec 15 '16 at 18:39
  • Exactly- if two shapes are congruent, then they're pretty much the same. Same angles, same lengths, etc. So if they're congruent, then everything about them is congruent, including their angles. Now you've proved one direction of this proof. Now assume that the central angles of these arcs are congruent. For the arcs themselves to be congruent, they'd need to have the same arc length. How do you measure arc length? – Kevin Long Dec 15 '16 at 18:44
  • When you divide the arc's degree by 360 degrees, there's the fraction of the circle's circumference the arc makes up and then multiply the length all the way around the circle and by this fraction, there's the length along the arc – Danneh Corn Dec 15 '16 at 18:53
  • Right. In other words, the arc length of an arc with angle $\theta$ in a circle of radius $r$ is $2\pi r\frac{\theta}{2\pi}$. So both arcs have angle $\theta$. Let's say that the first arc is in a circle $C_1$ of radius $r_1$ and the second arc is in a circle $C_2$ of radius $r_2$. Then if $r_1=r_2$, their arc lengths are equal. If they belong to the same circle, then of course $r_1=r_2$. What if $C_1$ and $C_2$ are congruent instead? – Kevin Long Dec 15 '16 at 18:59
  • C1 would equal C2 and the radius's would equal each other too? I'm not sure, i'm feeling really dumb here :/ – Danneh Corn Dec 15 '16 at 19:30
  • I'll put the rest as an answer, since comments aren't for continued discussion, and you don't have enough reputation to chat, but you have the right idea. – Kevin Long Dec 15 '16 at 19:44

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The "only if" direction of this proof is easy to show, since the congruence of two arcs means by definition that their angles are congruent.

The "if" direction requires you to show that their arc lengths are equal. Recall the formula for arclength: $2\pi r\frac{\theta}{2\pi}$ where $r$ is the radius of the circle containing the arc, and $\theta$ is the central angle of the arc. If the first arc belongs to circle $C_1$ with radius $r_1$ and the second belongs to circle $C_2$ with radius $r_2$, and their central angles are congruent, then you need only show that $r_1=r_2$ to get equality of the arc lengths, and hence, congruence of the arcs. If they belong to the same circle, this is obvious. If they belong to congruent circles, then again, by definition of congruence, their radii must be equal.

Kevin Long
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