{"id":1500,"date":"2024-02-06T22:50:58","date_gmt":"2024-02-06T17:50:58","guid":{"rendered":"https:\/\/rileymath.com\/?p=1500"},"modified":"2024-02-07T18:55:46","modified_gmt":"2024-02-07T13:55:46","slug":"qed-book-1-proposition-5","status":"publish","type":"post","link":"https:\/\/rileymath.com\/index.php\/2024\/02\/06\/qed-book-1-proposition-5\/","title":{"rendered":"QED – Book 1, Proposition 5"},"content":{"rendered":"\t\t
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Here we are in book 1 of Euclid’s Elements. \u00a0Just for fun, I decided to skip ahead a few to the first interesting proof in the Elements, which is Proposition 5 involving Isosceles triangles. I will come back to Proposition 2 through 4 later.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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Proposition 5<\/u><\/b><\/span><\/h3>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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In isosceles triangles, the angles at the base are equal to one another, and, if the equal straight lines be produced further, the angles under the base will be equal to one another.<\/span><\/strong><\/em><\/h3>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t
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Let\u00a0<\/span>ABC\u00a0<\/span>be an isosceles triangle having the side\u00a0<\/span>AB\u00a0<\/span>equal to the side\u00a0<\/span>AC<\/span>, and let the straight-lines\u00a0<\/span>BD\u00a0<\/span>and\u00a0<\/span>CE\u00a0<\/span>have been produced in a straight-line with\u00a0<\/span>AB\u00a0<\/span>and\u00a0<\/span>AC\u00a0<\/span>(respectively). I say that the angle\u00a0<\/span>ABC\u00a0<\/span>is equal to\u00a0<\/span>ACB<\/span>, and (angle)\u00a0<\/span>CBD\u00a0<\/span>to\u00a0<\/span>BCE<\/span>.<\/span><\/i><\/b><\/p>

I find it interesting that\u00a0<\/span>Euclid<\/span><\/span>\u00a0speaks in the first person in the proof. \u00a0Not that their is anything wrong with that. \u00a0It just seems informal for someone who is probably writing in the Library of Alexandria, Egypt (Pre-Fire).<\/span><\/p><\/div><\/div><\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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I have made markings in green to show the given information and the red marks are the angle which must be proven congruent.  (Euclid says equal, but we all know equal means same measure and same location, and we know he meant congruent which means “equal in measure”.  Another geometric term which has apparently evolved in 2500 years.)<\/p>\n

Another way to word this proof is:  The base angles of an Isosceles triangle are congruent.<\/p>\n

Back to Euclid:<\/p>\n

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For let the point <\/span>F <\/span>have been taken at random on <\/span>BD<\/span>, and let <\/span>AG <\/span>have been cut off from the greater <\/span>AE<\/span>, equal <\/span>to the lesser <\/span>AF<\/span>. Also, let the straight-lines <\/span>F C <\/span>and <\/span>GB <\/span>have been joined.  <\/b><\/i><\/span>In fact, since <\/span>AF <\/span>is equal to <\/span>AG<\/span>, and <\/span>AB <\/span>to <\/span>AC<\/span>, the two (straight-lines) <\/span>FA<\/span>, <\/span>AC <\/span>are equal to the two (straight-lines) <\/span>GA<\/span>, <\/span>AB<\/span>, respectively.  <\/span><\/b>They also encom- pass a common angle, <\/span>F AG<\/span>. Thus, the base <\/span>F C <\/span>is equal to the base <\/span>GB<\/span>, and the triangle <\/span>AF C <\/span>will be equal to the triangle <\/span>AGB<\/span>, and the remaining angles subtendend by the equal sides will be equal to the corresponding remain- ing angles<\/span><\/b><\/i><\/p>\n

It seems his strategy is to show that triangle FBC is congruent to triangle GCB (by side-side-side) and therefore the obtuse angles in the drawing FBC and GCB will be congruent and, by linear pairs, the bas angles of the original triangle will be congruent.  One more drawing.  Congruent sides are :<\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\"\"\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"

Here we are in book 1 of Euclid’s Elements. \u00a0Just for fun, I decided to skip ahead a few to the first interesting proof in the Elements, which is Proposition 5 involving Isosceles triangles. I will come back to Proposition 2 through 4 later. Proposition 5 In isosceles triangles, the angles at the base are […]<\/p>\n","protected":false},"author":1,"featured_media":1530,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","footnotes":""},"categories":[1],"tags":[],"yoast_head":"\nQED - Book 1, Proposition 5 - Riley Math Education<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/rileymath.com\/index.php\/2024\/02\/06\/qed-book-1-proposition-5\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"QED - Book 1, Proposition 5 - Riley Math Education\" \/>\n<meta property=\"og:description\" content=\"Here we are in book 1 of Euclid’s Elements. \u00a0Just for fun, I decided to skip ahead a few to the first interesting proof in the Elements, which is Proposition 5 involving Isosceles triangles. 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After ten years as an officer in the United States Navy and eight additional years in management, I decided to become a teacher of mathematics, physics and earth science. About ten years into my teaching career, I discovered that I was an ineffective teacher. I did some research and found that most other teachers were also ineffective. I did more research and changed my traditional teaching style to modern methods where students worked on projects in class and were allowed to be creative and thoughtful. I was now a more effective teacher, but I still had more research to do and I continue pursuing better teaching methods to this day. After two years of project-based teaching, I began visiting classrooms throughout central Indiana on my prep periods to run activities and talk about brain science and math methods to teachers. 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After ten years as an officer in the United States Navy and eight additional years in management, I decided to become a teacher of mathematics, physics and earth science. About ten years into my teaching career, I discovered that I was an ineffective teacher. I did some research and found that most other teachers were also ineffective. I did more research and changed my traditional teaching style to modern methods where students worked on projects in class and were allowed to be creative and thoughtful. I was now a more effective teacher, but I still had more research to do and I continue pursuing better teaching methods to this day. After two years of project-based teaching, I began visiting classrooms throughout central Indiana on my prep periods to run activities and talk about brain science and math methods to teachers. Now with 18 years as a teacher and having taught every math subject from 6th grade through AP and IB math classes, I am leaving the classroom to become a full time trainer. I will visit as many classrooms as I can for the rest of my life to pass on what I have learned.","sameAs":["https:\/\/www.rileymath.com"],"url":"https:\/\/rileymath.com\/index.php\/author\/john\/"}]}},"uagb_featured_image_src":{"full":["https:\/\/rileymath.com\/wp-content\/uploads\/2024\/02\/EUclid-1.5-final-image.jpeg",960,1280,false],"thumbnail":["https:\/\/rileymath.com\/wp-content\/uploads\/2024\/02\/EUclid-1.5-final-image-150x150.jpeg",150,150,true],"medium":["https:\/\/rileymath.com\/wp-content\/uploads\/2024\/02\/EUclid-1.5-final-image-225x300.jpeg",225,300,true],"medium_large":["https:\/\/rileymath.com\/wp-content\/uploads\/2024\/02\/EUclid-1.5-final-image-768x1024.jpeg",768,1024,true],"large":["https:\/\/rileymath.com\/wp-content\/uploads\/2024\/02\/EUclid-1.5-final-image-768x1024.jpeg",768,1024,true],"1536x1536":["https:\/\/rileymath.com\/wp-content\/uploads\/2024\/02\/EUclid-1.5-final-image.jpeg",960,1280,false],"2048x2048":["https:\/\/rileymath.com\/wp-content\/uploads\/2024\/02\/EUclid-1.5-final-image.jpeg",960,1280,false]},"uagb_author_info":{"display_name":"John Riley","author_link":"https:\/\/rileymath.com\/index.php\/author\/john\/"},"uagb_comment_info":0,"uagb_excerpt":"Here we are in book 1 of Euclid’s Elements. \u00a0Just for fun, I decided to skip ahead a few to the first interesting proof in the Elements, which is Proposition 5 involving Isosceles triangles. I will come back to Proposition 2 through 4 later. Proposition 5 In isosceles triangles, the angles at the base are…","_links":{"self":[{"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/posts\/1500"}],"collection":[{"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/comments?post=1500"}],"version-history":[{"count":14,"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/posts\/1500\/revisions"}],"predecessor-version":[{"id":1537,"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/posts\/1500\/revisions\/1537"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/media\/1530"}],"wp:attachment":[{"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/media?parent=1500"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/categories?post=1500"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rileymath.com\/index.php\/wp-json\/wp\/v2\/tags?post=1500"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}