We know that arcs that are equal in size and shape are called congruent arcs. My book defines the degree measure of an arc like this-The degree measure of an arc is the measures of the central angle containing the arc. Now,my question is that, How should we prove that two arcs of the same or congruent circles are congruent if they have the same degree measure?
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Do you know that any two angles of equal measure are congruent? – Eric Wofsey Nov 13 '16 at 04:49
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Arcs on same (or congruent) circles have the same "shape", and the "size" is measured by the angle, so it's not obvious what you mean to prove that doesn't come directly from the definitions. It would help if you pointed out where your difficulty is with the original question, and what the difference is that you see vs. the following paraphrase:
The distance between two points is defined as the measure of the segment between the two points. How should we prove that two segments are congruent if the distances between their respective endpoints are the same?
dxiv
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1Congruent circles have the same radius, thus the same curvature. Informally speaking, they have the same "shape" in a sense. If you cut two arcs from congruent circles, you can always fit the smaller one inside the larger one. To continue the analogy with straight lines, all segments have $0$ curvature and the same straight "shape". And you can always fit a smaller segment inside a larger one. – dxiv Nov 13 '16 at 17:38
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1) You divide the angle of both arcs into $n$ equal sub-angles and approximate the arc by an open polygon.
2) By the means of congruent triangles, you prove that the length of the approximating open polygons is the same.
3) Then when $n$ is large (in limit when $n\rightarrow\infty$) the open polygons will become the arcs. The more sub-angles you have, the better your open polygon will approximate the arc. As your open polygons have equal length, the arcs will be equal too
Momo
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I have not studied Calculus.is'n there any other simple geometrical proof? – CandidFlakes Nov 13 '16 at 03:13
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I mean one cannot prove curves congruence with triangle congruence unless some kind of approximation is made. – Momo Nov 13 '16 at 03:34
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@NavneetKumar It is sufficient show that the figures are congruent. By symmetry, it is sufficient to show that a rotation of the figure that shows arc A is the figure that shows arc B.
Jacob Wakem
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