Two arcs of a circle are congruent if and only if their associated radii are congruent

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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're congruent circles. If two arcs are both equal in measure and they're segments of congruent circles, then they're 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'. With biconditional statements, we can't always just reverse the argument to get the reverse implication, but in this case we can.

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.

{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T20:32:05+00:00","modifiedTime":"2016-03-26T20:32:05+00:00","timestamp":"2022-06-22T19:27:03+00:00"},"data":{"breadcrumbs":[{"name":"Academics & The Arts","_links":{"self":"//dummies-api.dummies.com/v2/categories/33662"},"slug":"academics-the-arts","categoryId":33662},{"name":"Math","_links":{"self":"//dummies-api.dummies.com/v2/categories/33720"},"slug":"math","categoryId":33720},{"name":"Geometry","_links":{"self":"//dummies-api.dummies.com/v2/categories/33725"},"slug":"geometry","categoryId":33725}],"title":"Six Important Circle Theorems","strippedTitle":"six important circle theorems","slug":"six-important-circle-theorems","canonicalUrl":"","seo":{"metaDescription":"The six circle theorems discussed here are all just variations on one basic idea about the interconnectedness of arcs, central angles, and chords (all six are i","noIndex":0,"noFollow":0},"content":"<p>The six circle theorems discussed here are all just variations on one basic idea about the interconnectedness of arcs, central angles, and chords (all six are illustrated in the following figure):</p>\n<img src=\"//sg.cdnki.com/two-arcs-of-a-circle-are-congruent-if-and-only-if-their-associated-radii-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI2MjAyMy5pbWFnZTAuanBn.webp\" width=\"375\" height=\"400\" alt=\"image0.jpg\"/>\n<h2 id=\"tab1\" >Central angles and arcs:</h2>\n<p><b>1.</b><b> If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent.</b> (Short form: If central angles congruent, then arcs congruent.) </p>\n<img src=\"//sg.cdnki.com/two-arcs-of-a-circle-are-congruent-if-and-only-if-their-associated-radii-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI2MjAyNC5pbWFnZTEucG5n.webp\" width=\"349\" height=\"27\" alt=\"image1.png\"/>\n<p><b>2.</b><b> If two arcs of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.</b> (Short form: If arcs congruent, then central angles congruent.) </p>\n<img src=\"//sg.cdnki.com/two-arcs-of-a-circle-are-congruent-if-and-only-if-their-associated-radii-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI2MjAyNS5pbWFnZTIucG5n.webp\" width=\"227\" height=\"25\" alt=\"image2.png\"/>\n<h2 id=\"tab2\" >Central angles and chords:</h2>\n<p><b>3.</b><b> If two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.</b> (Short form: If central angles congruent, then chords congruent.) </p>\n<img src=\"//sg.cdnki.com/two-arcs-of-a-circle-are-congruent-if-and-only-if-their-associated-radii-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI2MjAyNi5pbWFnZTMucG5n.webp\" width=\"352\" height=\"27\" alt=\"image3.png\"/>\n<p><b>4.</b><b> If two chords of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.</b> (Short form: If chords congruent, then central angles congruent.) </p>\n<img src=\"//sg.cdnki.com/two-arcs-of-a-circle-are-congruent-if-and-only-if-their-associated-radii-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI2MjAyNy5pbWFnZTQucG5n.webp\" width=\"227\" height=\"25\" alt=\"image4.png\"/>\n<h2 id=\"tab3\" >Arcs and chords:</h2>\n<p><b>5.</b><b> If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are cong</b><b>ruent.</b> (Short form: If arcs congruent, then chords congruent.) </p>\n<img src=\"//sg.cdnki.com/two-arcs-of-a-circle-are-congruent-if-and-only-if-their-associated-radii-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI2MjAyOC5pbWFnZTUucG5n.webp\" width=\"305\" height=\"27\" alt=\"image5.png\"/>\n<p><b>6.</b><b> If two chords of a circle (or of congruent circles) are congruent, then the corresponding arcs are co</b><b>ngruent.</b> (Short form: If chords congruent, then arcs congruent.) </p>\n<img src=\"//sg.cdnki.com/two-arcs-of-a-circle-are-congruent-if-and-only-if-their-associated-radii-are-congruent---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI2MjAyOS5pbWFnZTYucG5n.webp\" width=\"181\" height=\"25\" alt=\"image6.png\"/>\n<p>Here’s a more condensed way of thinking about the six theorems:</p>\n<ul class=\"level-one\">\n <li><p class=\"first-para\">If the angles are congruent, both the chords and the arcs are congruent.</p>\n </li>\n <li><p class=\"first-para\">If the chords are congruent, both the angles and the arcs are congruent.</p>\n </li>\n <li><p class=\"first-para\">If the arcs are congruent, both the angles and the chords are congruent.</p>\n </li>\n</ul>\n<p>These three ideas condense further to one simple idea: If any pair (of central angles, chords, or arcs) is congruent, then the other two pairs are also congruent.</p>","description":"<p>The six circle theorems discussed here are all just variations on one basic idea about the interconnectedness of arcs, central angles, and chords (all six are illustrated in the following figure):</p>\n<img src=\"//www.dummies.com/wp-content/uploads/262023.image0.jpg\" width=\"375\" height=\"400\" alt=\"image0.jpg\"/>\n<h2 id=\"tab1\" >Central angles and arcs:</h2>\n<p><b>1.</b><b> If two central angles of a circle (or of congruent circles) are congruent, then their intercepted arcs are congruent.</b> (Short form: If central angles congruent, then arcs congruent.) </p>\n<img src=\"//www.dummies.com/wp-content/uploads/262024.image1.png\" width=\"349\" height=\"27\" alt=\"image1.png\"/>\n<p><b>2.</b><b> If two arcs of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.</b> (Short form: If arcs congruent, then central angles congruent.) </p>\n<img src=\"//www.dummies.com/wp-content/uploads/262025.image2.png\" width=\"227\" height=\"25\" alt=\"image2.png\"/>\n<h2 id=\"tab2\" >Central angles and chords:</h2>\n<p><b>3.</b><b> If two central angles of a circle (or of congruent circles) are congruent, then the corresponding chords are congruent.</b> (Short form: If central angles congruent, then chords congruent.) </p>\n<img src=\"//www.dummies.com/wp-content/uploads/262026.image3.png\" width=\"352\" height=\"27\" alt=\"image3.png\"/>\n<p><b>4.</b><b> If two chords of a circle (or of congruent circles) are congruent, then the corresponding central angles are congruent.</b> (Short form: If chords congruent, then central angles congruent.) </p>\n<img src=\"//www.dummies.com/wp-content/uploads/262027.image4.png\" width=\"227\" height=\"25\" alt=\"image4.png\"/>\n<h2 id=\"tab3\" >Arcs and chords:</h2>\n<p><b>5.</b><b> If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are cong</b><b>ruent.</b> (Short form: If arcs congruent, then chords congruent.) </p>\n<img src=\"//www.dummies.com/wp-content/uploads/262028.image5.png\" width=\"305\" height=\"27\" alt=\"image5.png\"/>\n<p><b>6.</b><b> If two chords of a circle (or of congruent circles) are congruent, then the corresponding arcs are co</b><b>ngruent.</b> (Short form: If chords congruent, then arcs congruent.) </p>\n<img src=\"//www.dummies.com/wp-content/uploads/262029.image6.png\" width=\"181\" height=\"25\" alt=\"image6.png\"/>\n<p>Here’s a more condensed way of thinking about the six theorems:</p>\n<ul class=\"level-one\">\n <li><p class=\"first-para\">If the angles are congruent, both the chords and the arcs are congruent.</p>\n </li>\n <li><p class=\"first-para\">If the chords are congruent, both the angles and the arcs are congruent.</p>\n </li>\n <li><p class=\"first-para\">If the arcs are congruent, both the angles and the chords are congruent.</p>\n </li>\n</ul>\n<p>These three ideas condense further to one simple idea: If any pair (of central angles, chords, or arcs) is congruent, then the other two pairs are also congruent.</p>","blurb":"","authors":[{"authorId":8957,"name":"Mark Ryan","slug":"mark-ryan","description":" <p><b>Mark Ryan</b> is the owner of The Math Center in Chicago, Illinois, where he teaches students in all levels of mathematics, from pre&#45;algebra to calculus. He is the author of <i>Calculus For Dummies</i> and <i> Geometry For Dummies.</i> ","_links":{"self":"//dummies-api.dummies.com/v2/authors/8957"}}],"primaryCategoryTaxonomy":{"categoryId":33725,"title":"Geometry","slug":"geometry","_links":{"self":"//dummies-api.dummies.com/v2/categories/33725"}},"secondaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"tertiaryCategoryTaxonomy":{"categoryId":0,"title":null,"slug":null,"_links":null},"trendingArticles":null,"inThisArticle":[{"label":"Central angles and arcs:","target":"#tab1"},{"label":"Central angles and chords:","target":"#tab2"},{"label":"Arcs and chords:","target":"#tab3"}],"relatedArticles":{"fromBook":[],"fromCategory":[{"articleId":230077,"title":"How to Copy an Angle Using a Compass","slug":"copy-angle-using-compass","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230077"}},{"articleId":230072,"title":"How to Copy a Line Segment Using a Compass","slug":"copy-line-segment-using-compass","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230072"}},{"articleId":230069,"title":"How to Find the Right Angle to Two Points","slug":"find-right-angle-two-points","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230069"}},{"articleId":230066,"title":"Find the Locus of Points Equidistant from Two Points","slug":"find-locus-points-equidistant-two-points","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230066"}},{"articleId":230063,"title":"How to Solve a Two-Dimensional Locus Problem","slug":"solve-two-dimensional-locus-problem","categoryList":["academics-the-arts","math","geometry"],"_links":{"self":"//dummies-api.dummies.com/v2/articles/230063"}}]},"hasRelatedBookFromSearch":true,"relatedBook":{"bookId":282230,"slug":"geometry-for-dummies-3rd-edition","isbn":"9781119181552","categoryList":["academics-the-arts","math","geometry"],"amazon":{"default":"//www.amazon.com/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20","ca":"//www.amazon.ca/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20","indigo_ca":"//www.tkqlhce.com/click-9208661-13710633?url=//www.chapters.indigo.ca/en-ca/books/product/1119181550-item.html&cjsku=978111945484","gb":"//www.amazon.co.uk/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20","de":"//www.amazon.de/gp/product/1119181550/ref=as_li_tl?ie=UTF8&tag=wiley01-20"},"image":{"src":"//catalogimages.wiley.com/images/db/jimages/9781119181552.jpg","width":250,"height":350},"title":"Geometry For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"\n <p><p><b><b data-author-id=\"8957\">Mark Ryan</b> </b>is the founder and owner of The Math Center in the Chicago area, where he provides tutoring in all math subjects as well as test preparation. 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