Learn about the properties of isosceles trapezoids including relationships among opposite sides, opposite angles, adjacent angles, diagonals and angles formed by diagonals.
Isosceles Trapezoid: A trapezoid with one pair of parallel lines and the non-parallel sides congruent.
Isosceles trapezoids are special types of trapezoids that have the pair of of non-parallel legs being congruent to each other. This means that the trapezoid appears symmetrical, and that the diagonals are equal in length.
Like an isosceles triangle, isosceles trapezoids have base angles that are congruent. This means that the two smaller angles are congruent to each other, and the two larger angles are congruent to each other.
When diagonals are drawn, the still do not bisect each other. The bottom part of the two diagonals are congruent to each other, and the top part of the two diagonals are also congruent to each other.
An isosceles trapezoid also has two of the opposite triangles formed by the diagonals that are similar to each other, meaning all their sides and angles are in proportion. The other two opposite triangles formed are congruent to each other by side-side-side.
{"appState":{"pageLoadApiCallsStatus":true},"articleState":{"article":{"headers":{"creationTime":"2016-03-26T20:38:11+00:00","modifiedTime":"2021-07-09T13:59:53+00:00","timestamp":"2022-09-14T18:18:23+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":"The Properties of Trapezoids and Isosceles Trapezoids","strippedTitle":"the properties of trapezoids and isosceles trapezoids","slug":"the-properties-of-trapezoids-and-isosceles-trapezoids","canonicalUrl":"","seo":{"metaDescription":"Learn about two different types of trapezoids and the properties that define them, including parallel bases and supplementary angles.","noIndex":0,"noFollow":0},"content":"A trapezoid is a quadrilateral (a shape with four sides) with exactly one pair of parallel sides (the parallel sides are called <i>bases</i>). The following figure shows a trapezoid to the left, and an isosceles trapezoid on the right.\r\n\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYyMy5pbWFnZTAuanBn.webp\" alt=\"image0.jpg\" width=\"535\" height=\"112\" />\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The properties of the trapezoid are as follows:</p>\r\n\r\n<ul class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">The bases are parallel by definition.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Each lower base angle is supplementary to the upper base angle on the same side.</p>\r\n</li>\r\n</ul>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The properties of the isosceles trapezoid are as follows:</p>\r\n\r\n<ul class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">The properties of a trapezoid apply by definition (parallel bases).</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The legs are congruent by definition.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The lower base angles are congruent.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The upper base angles are congruent.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Any lower base angle is supplementary to any upper base angle.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The diagonals are congruent.</p>\r\n</li>\r\n</ul>\r\n</li>\r\n</ul>\r\nThe supplementary angles might be the hardest property to spot in the diagrams above. Because of the parallel sides, consecutive angles are same-side interior angles and are thus supplementary. (All the special quadrilaterals except the kite, by the way, contain consecutive supplementary angles.)\r\n\r\nHere’s an isosceles trapezoid proof for you:\r\n\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYyNC5pbWFnZTEucG5n.webp\" alt=\"image1.png\" width=\"395\" height=\"48\" />\r\n\r\n<b>Statement 1</b><b>:</b>\r\n\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYyNS5pbWFnZTIuanBn.webp\" alt=\"image2.jpg\" width=\"467\" height=\"336\" />\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYyNi5pbWFnZTMucG5n.webp\" alt=\"image3.png\" width=\"345\" height=\"27\" />\r\n\r\n<i>Reason for statement 1</i><i>:</i> Given.\r\n\r\n<b>Statement 2</b><b>:</b>\r\n\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYyNy5pbWFnZTQucG5n.webp\" alt=\"image4.png\" width=\"56\" height=\"24\" />\r\n\r\n<i>Reason for statement 2</i><i>:</i> The legs of an isosceles trapezoid are congruent.\r\n\r\n<b>Statement 3</b><b>:</b>\r\n\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYyOC5pbWFnZTUucG5n.webp\" alt=\"image5.png\" width=\"95\" height=\"19\" />\r\n\r\n<i>Reason for statement 3</i><i>:</i> The upper base angles of an isosceles trapezoid are congruent.\r\n\r\n<b>Statement 4</b><b>:</b>\r\n\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYyOS5pbWFnZTYucG5n.webp\" alt=\"image6.png\" width=\"51\" height=\"24\" />\r\n\r\n<i>Reason for statement 4</i><i>:</i> Reflexive Property.\r\n\r\n<b>Statement 5</b><b>:</b>\r\n\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYzMC5pbWFnZTcucG5n.webp\" alt=\"image7.png\" width=\"84\" height=\"19\" />\r\n\r\n<i>Reason for statement 5</i><i>:</i> Side-Angle-Side, or SAS (2, 3, 4)\r\n\r\n<b>Statement 6</b><b>:</b>\r\n\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYzMS5pbWFnZTgucG5n.webp\" alt=\"image8.png\" width=\"95\" height=\"19\" />\r\n\r\n<i>Reason for statement 6</i><i>:</i> CPCTC (Corresponding Parts of Congruent Triangles are Congruent).\r\n\r\n<b>Statement 7</b><b>:</b>\r\n\r\n<img src=\"//sg.cdnki.com/how-long-are-the-opposite-angles-of-an-isosceles-trapezoid-related---aHR0cHM6Ly93d3cuZHVtbWllcy5jb20vd3AtY29udGVudC91cGxvYWRzLzI1ODYzMi5pbWFnZTkucG5n.webp\" alt=\"image9.png\" width=\"52\" height=\"24\" />\r\n\r\n<i>Reason for statement 7</i><i>:</i><i> </i>If angles are congruent, then so are sides.","description":"A trapezoid is a quadrilateral (a shape with four sides) with exactly one pair of parallel sides (the parallel sides are called <i>bases</i>). The following figure shows a trapezoid to the left, and an isosceles trapezoid on the right.\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/258623.image0.jpg\" alt=\"image0.jpg\" width=\"535\" height=\"112\" />\r\n<ul class=\"level-one\">\r\n \t<li>\r\n<p class=\"first-para\">The properties of the trapezoid are as follows:</p>\r\n\r\n<ul class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">The bases are parallel by definition.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Each lower base angle is supplementary to the upper base angle on the same side.</p>\r\n</li>\r\n</ul>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The properties of the isosceles trapezoid are as follows:</p>\r\n\r\n<ul class=\"level-two\">\r\n \t<li>\r\n<p class=\"first-para\">The properties of a trapezoid apply by definition (parallel bases).</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The legs are congruent by definition.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The lower base angles are congruent.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The upper base angles are congruent.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">Any lower base angle is supplementary to any upper base angle.</p>\r\n</li>\r\n \t<li>\r\n<p class=\"first-para\">The diagonals are congruent.</p>\r\n</li>\r\n</ul>\r\n</li>\r\n</ul>\r\nThe supplementary angles might be the hardest property to spot in the diagrams above. Because of the parallel sides, consecutive angles are same-side interior angles and are thus supplementary. (All the special quadrilaterals except the kite, by the way, contain consecutive supplementary angles.)\r\n\r\nHere’s an isosceles trapezoid proof for you:\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/258624.image1.png\" alt=\"image1.png\" width=\"395\" height=\"48\" />\r\n\r\n<b>Statement 1</b><b>:</b>\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/258625.image2.jpg\" alt=\"image2.jpg\" width=\"467\" height=\"336\" />\r\n<img src=\"//www.dummies.com/wp-content/uploads/258626.image3.png\" alt=\"image3.png\" width=\"345\" height=\"27\" />\r\n\r\n<i>Reason for statement 1</i><i>:</i> Given.\r\n\r\n<b>Statement 2</b><b>:</b>\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/258627.image4.png\" alt=\"image4.png\" width=\"56\" height=\"24\" />\r\n\r\n<i>Reason for statement 2</i><i>:</i> The legs of an isosceles trapezoid are congruent.\r\n\r\n<b>Statement 3</b><b>:</b>\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/258628.image5.png\" alt=\"image5.png\" width=\"95\" height=\"19\" />\r\n\r\n<i>Reason for statement 3</i><i>:</i> The upper base angles of an isosceles trapezoid are congruent.\r\n\r\n<b>Statement 4</b><b>:</b>\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/258629.image6.png\" alt=\"image6.png\" width=\"51\" height=\"24\" />\r\n\r\n<i>Reason for statement 4</i><i>:</i> Reflexive Property.\r\n\r\n<b>Statement 5</b><b>:</b>\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/258630.image7.png\" alt=\"image7.png\" width=\"84\" height=\"19\" />\r\n\r\n<i>Reason for statement 5</i><i>:</i> Side-Angle-Side, or SAS (2, 3, 4)\r\n\r\n<b>Statement 6</b><b>:</b>\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/258631.image8.png\" alt=\"image8.png\" width=\"95\" height=\"19\" />\r\n\r\n<i>Reason for statement 6</i><i>:</i> CPCTC (Corresponding Parts of Congruent Triangles are Congruent).\r\n\r\n<b>Statement 7</b><b>:</b>\r\n\r\n<img src=\"//www.dummies.com/wp-content/uploads/258632.image9.png\" alt=\"image9.png\" width=\"52\" height=\"24\" />\r\n\r\n<i>Reason for statement 7</i><i>:</i><i> </i>If angles are congruent, then so are sides.","blurb":"","authors":[{"authorId":8957,"name":"Mark Ryan","slug":"mark-ryan","description":" <p><b>Mark Ryan</b> has taught pre-algebra through calculus for more than 25 years. In 1997, he founded The Math Center in Winnetka, Illinois, where he teaches junior high and high school mathematics courses as well as standardized test prep classes. He also does extensive one-on-one tutoring. He is a member of the Authors Guild and the National Council of Teachers of Mathematics. ","hasArticle":false,"_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":[],"relatedArticles":{"fromBook":[{"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"}}],"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":false,"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":"//www.dummies.com/wp-content/uploads/geometry-for-dummies-3rd-edition-cover-9781119181552-201x255.jpg","width":201,"height":255},"title":"Geometry For Dummies","testBankPinActivationLink":"","bookOutOfPrint":false,"authorsInfo":"<p><b data-author-id=\"8957\">Mark Ryan </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. Mark is the author of <i>Calculus For Dummies, Calculus Workbook For Dummies</i>, and <i>Geometry Workbook For Dummies</i>.</p>","authors":[{"authorId":8957,"name":"Mark Ryan","slug":"mark-ryan","description":" <p><b>Mark Ryan</b> has taught pre-algebra through calculus for more than 25 years. In 1997, he founded The Math Center in Winnetka, Illinois, where he teaches junior high and high school mathematics courses as well as standardized test prep classes. He also does extensive one-on-one tutoring. He is a member of the Authors Guild and the National Council of Teachers of Mathematics. ","hasArticle":false,"_links":{"self":"//dummies-api.dummies.com/v2/authors/8957"}}],"_links":{"self":"//dummies-api.dummies.com/v2/books/"}},"collections":[],"articleAds":{"footerAd":"<div class=\"du-ad-region row\" id=\"article_page_adhesion_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_adhesion_ad\" data-refreshed=\"false\" \r\n data-target = \"[{"key":"cat","values":["academics-the-arts","math","geometry"]},{"key":"isbn","values":["9781119181552"]}]\" id=\"du-slot-63221aefcff4e\"></div></div>","rightAd":"<div class=\"du-ad-region row\" id=\"article_page_right_ad\"><div class=\"du-ad-unit col-md-12\" data-slot-id=\"article_page_right_ad\" data-refreshed=\"false\" \r\n data-target = \"[{"key":"cat","values":["academics-the-arts","math","geometry"]},{"key":"isbn","values":["9781119181552"]}]\" 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The properties of the trapezoid are as follows:
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The bases are parallel by definition.
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Each lower base angle is supplementary to the upper base angle on the same side.
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The properties of the isosceles trapezoid are as follows:
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The properties of a trapezoid apply by definition (parallel bases).
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The legs are congruent by definition.
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The lower base angles are congruent.
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The upper base angles are congruent.
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Any lower base angle is supplementary to any upper base angle.
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The diagonals are congruent.
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Here’s an isosceles trapezoid proof for you:
Statement 1:
Reason for statement 1: Given.
Statement 2:
Reason for statement 2: The legs of an isosceles trapezoid are congruent.
Statement 3:
Reason for statement 3: The upper base angles of an isosceles trapezoid are congruent.
Statement 4:
Reason for statement 4: Reflexive Property.
Statement 5:
Reason for statement 5: Side-Angle-Side, or SAS (2, 3, 4)
Statement 6:
Reason for statement 6: CPCTC (Corresponding Parts of Congruent Triangles are Congruent).
Statement 7:
Reason for statement 7: If angles are congruent, then so are sides.