{"id":666,"date":"2013-05-28T15:04:18","date_gmt":"2013-05-28T15:04:18","guid":{"rendered":"http:\/\/www.mometrix.com\/academy\/?page_id=666"},"modified":"2026-03-26T09:58:16","modified_gmt":"2026-03-26T14:58:16","slug":"geometry-application-similar-triangles","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/geometry-application-similar-triangles\/","title":{"rendered":"Similar Triangles"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_4kmGoUXwIpo\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_4kmGoUXwIpo\" data-source-videoID=\"4kmGoUXwIpo\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Similar Triangles Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Similar Triangles\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_4kmGoUXwIpo:hover {cursor:pointer;} img#videoThumbnailImage_4kmGoUXwIpo {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1126-similar-triangles-1-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_4kmGoUXwIpo\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_4kmGoUXwIpo\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Similar Triangles\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_4kmGoUXwIpo\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_4kmGoUXwIpo\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_4kmGoUXwIpo\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction MOL_Function() {\n  var x = document.getElementById(\"MOL\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"MOL_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"MOL\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#Similar_Triangles_Definition\" class=\"smooth-scroll\">Similar Triangles Definition<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#AngleAngleAngle_Method\" class=\"smooth-scroll\">Angle-Angle-Angle Method<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#SideSideSide_Method\" class=\"smooth-scroll\">Side-Side-Side Method<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#SideAngleSide_Method\" class=\"smooth-scroll\">Side-Angle-Side Method<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Height_of_the_Tree_Problem\" class=\"smooth-scroll\">Height of the Tree Problem<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Review\" class=\"smooth-scroll\">Review<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Frequently_Asked_Questions\" class=\"smooth-scroll\">Frequently Asked Questions<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Similar_Triangle_Practice_Questions\" class=\"smooth-scroll\">Similar Triangle Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"FAQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"FAQs\">FAQs<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this review on similar triangles! Today, we\u2019re going to explore how to identify similar triangles and how we can use that knowledge to solve a very popular kind of geometry problem. <\/p>\n<h2><span id=\"Similar_Triangles_Definition\" class=\"m-toc-anchor\"><\/span>Similar Triangles Definition<\/h2>\n<p>\nLet\u2019s start with a simple definition. Similar triangles are triangles that have the same shape. That means they will have the same three <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/angles\/\">angles<\/a>. But here\u2019s the twist\u2014similar triangles don\u2019t have to be the same size! So if you take a copy of the triangle and dilate it to twice its size, it will still be similar to the original triangle.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/02\/Dilated-Triangles-02.svg\" alt=\"Dilated triangles\" width=\"408\" height=\"336\" class=\"aligncenter size-full wp-image-212659\"  role=\"img\" \/><\/p>\n<h2><span id=\"AngleAngleAngle_Method\" class=\"m-toc-anchor\"><\/span>Angle-Angle-Angle Method<\/h2>\n<p>\nNotice that the two triangles have the same shape and proportions. That\u2019s because they have the exact same angles, which is what makes them similar triangles. This is the <strong>Angle-Angle-Angle<\/strong>, or AAA, method of determining similarity. It should be noted that anytime you dilate a triangle, you end up with two similar triangles.<\/p>\n<p>Since we know these two triangles are similar triangles, we can express this in math notation like this: \\(\u25b3ABC\\) ~ \\(\u25b3A&#8217;B&#8217;C&#8217;\\) <\/p>\n<p>Okay, but what if we didn\u2019t start by dilating and don\u2019t know all the angles? Is there any other way to tell if two triangles are similar? <\/p>\n<h2><span id=\"SideSideSide_Method\" class=\"m-toc-anchor\"><\/span>Side-Side-Side Method<\/h2>\n<p>\nLet\u2019s look at an example where we only know the lengths of the sides of two triangles:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/two-triangles-example.png\" alt=\"Two triangles that are the same shape, but one is twice as large as the one next to it.\" width=\"600\" height=\"396.576\" class=\"aligncenter size-full wp-image-92020\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/two-triangles-example.png 1103w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/two-triangles-example-300x198.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/two-triangles-example-1024x677.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/two-triangles-example-768x508.png 768w\" sizes=\"(max-width: 1103px) 100vw, 1103px\" \/><\/p>\n<p>These two triangles appear to be the same shape, but remember that in geometry we can\u2019t always trust our eyes. We need to prove that they are the same shape, and we can do that by checking to see if their sides are in proportion to each other. <\/p>\n<p>We set up our proportions by setting ratios, or fractions, of the corresponding sides and setting them equal to each other, like this: We\u2019ll have the side lengths for \\(\u25b3ABC\\) be the numerators, and the lengths for \\(\u25b3DEF\\) be the denominators. It would work just as well if we switched it around.<\/p>\n<p>It\u2019s very important to label proportions to make sure we put all the numbers in their proper place.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/labeling-proportions.png\" alt=\"\" width=\"450\" height=\"355.9731\" class=\"aligncenter size-full wp-image-92026\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/labeling-proportions.png 852w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/labeling-proportions-300x237.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/labeling-proportions-768x608.png 768w\" sizes=\"(max-width: 852px) 100vw, 852px\" \/><\/p>\n<p>So what does this proportion do for us? Well, we need to test to see if it\u2019s true. In this case, we can do that by <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/multiplying-and-dividing-fractions\/\">reducing<\/a> each of these fractions to their simplest form. <\/p>\n<p>We can see that 3 goes into the top and bottom of \\(\\frac{6}{9}\\), so we can reduce it to 2 over 3. Four goes into \\(\\frac{8}{12}\\) and 5 goes into \\(\\frac{10}{15}\\). When we reduce those two fractions, we also end up with \\(\\frac{2}{3}\\). In reduced form our proportion looks like this: <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/labeling-proportions-2.png\" alt=\"\" width=\"450\" height=\"357.2382\" class=\"aligncenter size-full wp-image-92029\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/labeling-proportions-2.png 854w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/labeling-proportions-2-300x238.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/labeling-proportions-2-768x610.png 768w\" sizes=\"(max-width: 854px) 100vw, 854px\" \/><\/p>\n<p>This is proof that all of our sides are in proportion. If all the sides are in proportion, then we know that our triangles are in proportion. This is the <strong>Side-Side-Side<\/strong>, or SSS, method of proving similarity. So, now we can say that \\(\u25b3ABC\\) ~ \\(\u25b3DEF\\).<\/p>\n<p>Instead of reducing, we could also check if to see if the ratios are in proportion by converting each ratio to a decimal by dividing the top by the bottom of each. In this case, \\(\\frac{6}{9}\\), \\(\\frac{8}{12}\\), and \\(\\frac{10}{15}\\) would have given us the same decimal value, which is 0.66666666.<\/p>\n<p>So, now we know that if all the angles are the same, or if all the sides are in proportion, our two triangles are similar. <\/p>\n<h2><span id=\"SideAngleSide_Method\" class=\"m-toc-anchor\"><\/span>Side-Angle-Side Method<\/h2>\n<p>\nThere is one more method of proving similarity called <strong>Side-Angle-Side<\/strong>, or SAS. This is where we know the two triangles have one angle that is the same measure and that the two sides coming off that angle are in proportion.<\/p>\n<p>Let\u2019s take a look at our last two triangles, but instead of knowing all three sides, we only know two sets of corresponding sides along with the angle between them, like this: <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/similar-triangles-example-2.png\" alt=\"Two triangles that are the same shape, but one is twice as large as the one next to it.\" width=\"600\" height=\"372.6448\" class=\"aligncenter size-full wp-image-92032\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/similar-triangles-example-2.png 1098w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/similar-triangles-example-2-300x186.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/similar-triangles-example-2-1024x636.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/09\/similar-triangles-example-2-768x477.png 768w\" sizes=\"(max-width: 1098px) 100vw, 1098px\" \/><\/p>\n<p>We can see that the angle between the two measured sides is labeled with a square, which we know means that it\u2019s a right angle and therefore 90 degrees. Since that angle is the same in both triangles we just need to check to see if the two adjacent sides are in proportion, like this: <\/p>\n<p>Reducing the fractions again results in them being 2 over 3, so the sides are in proportion. Since the angle between those sides is the same, or <strong>congruent<\/strong>, this triangle is similar. <\/p>\n<h2><span id=\"Height_of_the_Tree_Problem\" class=\"m-toc-anchor\"><\/span>Height of the Tree Problem<\/h2>\n<p>\nOkay, so what can we do with this ability to recognize similar triangles? You\u2019ve may have seen it before, but here is the classic height of the tree problem: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-example-min.png\" alt=\"\" width=\"1492\" height=\"975\" class=\"aligncenter size-full wp-image-208973\" style=\"box-shadow: 1.5px 1.5px 3px grey;\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-example-min.png 1492w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-example-min-300x196.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-example-min-1024x669.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-example-min-768x502.png 768w\" sizes=\"auto, (max-width: 1492px) 100vw, 1492px\" \/><\/p>\n<p>Here\u2019s what we know. The tree and the flagstick are both standing straight up perpendicular to the ground. They are each casting a shadow from the same light source, the sun. The rays coming from the sun are straight lines shining on the objects at an angle. The ground is flat and can be considered a flat plane, which means any two points on it can be connected with a straight line. That means I can draw two triangles over this drawing, like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-method-triangles-min.png\" alt=\"\" width=\"1469\" height=\"962\" class=\"aligncenter size-full wp-image-208976\" style=\"box-shadow: 1.5px 1.5px 3px grey;\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-method-triangles-min.png 1469w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-method-triangles-min-300x196.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-method-triangles-min-1024x671.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/tree-method-triangles-min-768x503.png 768w\" sizes=\"auto, (max-width: 1469px) 100vw, 1469px\" \/><\/p>\n<p>Now, let\u2019s keep our triangles but lose the pretty scenery.<\/p>\n<p>Next, let\u2019s label all the points so we can reference them. It doesn\u2019t matter what letter we use for each point. <\/p>\n<p>We also know that angle B and angle D are right angles since the flagstick and the tree are pointing straight up perpendicular to the ground, so we\u2019ll label that too:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/similar-angles-in-triangles-min.png\" alt=\"\" width=\"1180\" height=\"806\" class=\"aligncenter size-full wp-image-208970\" style=\"box-shadow: 1.5px 1.5px 3px grey;\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/similar-angles-in-triangles-min.png 1180w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/similar-angles-in-triangles-min-300x205.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/similar-angles-in-triangles-min-1024x699.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/similar-angles-in-triangles-min-768x525.png 768w\" sizes=\"auto, (max-width: 1180px) 100vw, 1180px\" \/><\/p>\n<p>Since we know that angle A is the same for both triangles and that angles B and D are both right angles and therefore congruent, that means that angles C and E are also congruent. Why? Because the interior angles of a triangle must add up to 180\u00b0. So if A is 30\u00b0 and B and D are 90\u00b0, then C and E will end up being 60\u00b0. <\/p>\n<p>Now we know they are similar triangles using the AAA method of proving similarity. <\/p>\n<p>Now we can get to the fun part and use these triangles to find the height of the tree. First, we\u2019ll need to take some measurements. We need to measure the flagstick, the length of the shadow of the flagstick, and the length of the shadow of the tree. Fortunately, these are all easy to do. A standard golf flagstick is 7 feet tall. The shadows are on the ground, obviously, so we can measure them with a long tape measure. Then we can add them to our triangles. <\/p>\n<p>Since the sides of similar triangles are always in proportion, we can set up a proportion with all rows and columns labeled: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/height-of-flag-1536x865-min.png\" alt=\"\" width=\"1536\" height=\"865\" class=\"aligncenter size-full wp-image-208967\" style=\"box-shadow: 1.5px 1.5px 3px grey;\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/height-of-flag-1536x865-min.png 1536w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/height-of-flag-1536x865-min-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/height-of-flag-1536x865-min-1024x577.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/height-of-flag-1536x865-min-768x433.png 768w\" sizes=\"auto, (max-width: 1536px) 100vw, 1536px\" \/><\/p>\n<p>We don\u2019t know either of the diagonal sides but we don&#8217;t need them for this type of problem. We know the ground sides of both triangles and we know the height of \\(\u25b3ABC\\). We can place a variable for the height of the tree, which is \\(\\overline{DE}\\) in our triangle. Now we just use cross products to solve: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/cross-diagonal-1536x867-min.png\" alt=\"tree diagram\" width=\"1536\" height=\"867\" class=\"aligncenter size-full wp-image-208964\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/cross-diagonal-1536x867-min.png 1536w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/cross-diagonal-1536x867-min-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/cross-diagonal-1536x867-min-1024x578.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/12\/cross-diagonal-1536x867-min-768x434.png 768w\" sizes=\"auto, (max-width: 1536px) 100vw, 1536px\" \/><\/p>\n<p>Taking cross products results in the equation \\(9x=45\\cdot 7\\), or \\(9x=315\\) after we multiply 45 and 7. Dividing both sides by 9 gives us \\(x=35\\). So our tree is 35 feet tall! <\/p>\n<hr>\n<h2><span id=\"Review\" class=\"m-toc-anchor\"><\/span>Review<\/h2>\n<p>\nOk, now that we\u2019ve covered everything, let\u2019s do a quick recap:<\/p>\n<p>Similar triangles are triangles that have the same shape. There are three ways to prove that two triangles are similar. The AAA method of similarity is when the proportions of all three angles of the triangles are the same. The SSS method is when all three sides of the triangles are the same length. And finally, the SAS method is when the two triangles have one angle that is the same measure and the two sides coming off that angle are in proportion.<\/p>\n<p>I hope this review was helpful! Thanks for watching, and happy studying!<\/p>\n<ul class=\"citelist\">\n<li><a href=\"https:\/\/www.mathopenref.com\/similartriangles.html\"target=\"_blank\">\u201cSimilar Triangles &#8211; Math Open Reference.\u201d 2011. Mathopenref.com<\/a><\/li>\n<\/ul>\n<\/div>\n<div class=\"spoiler\" id=\"FAQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Frequently_Asked_Questions\" class=\"m-toc-anchor\"><\/span>Frequently Asked Questions<\/h2>\n<div class=\"faq-list\">\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">What are similar triangles?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Similar triangles are triangles whose corresponding angles are equal and whose corresponding sides are proportional but not necessarily equal.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Similar-Triangles-Example.svg\" alt=\"Two triangles are shown: triangle ABC (sides 3, 3, 6; angles 30\u00b0, 120\u00b0, 30\u00b0) and triangle DEF (sides 12, 12, 24; angles 30\u00b0, 120\u00b0, 30\u00b0).\" width=\"774\" height=\"161\" class=\"aligncenter size-full wp-image-273634\"  role=\"img\" \/><\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">Are all right triangles similar?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>No, not all right triangles are similar. For triangles to be similar, they must have the same angle measures. All right triangles have one right angle, but the other two angles can be any combination of measures that add to 90\u00b0.<\/p>\n<p>For example, \\(\\triangle\\)ABC is not similar to \\(\\triangle\\)DEF:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Right-triangles-example.svg\" alt=\"Two right triangles are shown; triangle ABC with two 45\u00b0 angles, and triangle DEF with a 30\u00b0 angle and a right angle.\" width=\"411\" height=\"222\" class=\"aligncenter size-full wp-image-273637\"  role=\"img\" \/><\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">Are all equilateral triangles similar?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>Yes, all equilateral triangles are similar because they all have the same angle measures.<\/p>\n<\/p><\/div>\n<\/p><\/div>\n<div class=\"qa_wrap\">\n<div class=\"q_item text_bold\">\n<h4 class=\"letter\">Q<\/h4>\n<p style=\"line-height: unset;\">Are all isoceles triangles similar?<\/p>\n<\/p><\/div>\n<div class=\"a_item\">\n<h4 class=\"letter text_bold\">A<\/h4>\n<p>No, not all isosceles triangles are similar. In order for triangles to be similar, they must have the same angle measures. Not all isosceles triangles have the same angle measures.<\/p>\n<p>For example, \\(\\triangle\\)ABC is not similar to \\(\\triangle\\)DEF:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Nonsimilar-triangles-example.svg\" alt=\"Two triangles: triangle ABC with angles 30\u00b0, 75\u00b0, and 75\u00b0, and triangle DEF with angles 100\u00b0, 40\u00b0, and 40\u00b0. Each triangle is labeled at its vertices.\" width=\"481\" height=\"214\" class=\"aligncenter size-full wp-image-273643\"  role=\"img\" \/><\/p>\n<\/p><\/div>\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Similar_Triangle_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Similar Triangle Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nWhich of the following statements is not true about similar triangles?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">They have the same shape<\/div><div class=\"PQ\"  id=\"PQ-1-2\">They have the same angle measures<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\">They are the same size<\/div><div class=\"PQ\"  id=\"PQ-1-4\">They have the same proportions<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Similar triangles are triangles that have the same shape, same angle measures, and are proportional to one another. Similar triangles are not the same size. If triangles are the same size, then they are congruent.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\n\\(\\triangle\\)ABC is similar to \\(\\triangle\\)DEF. Find \\(x\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Similar-triangles-example-3.svg\" alt=\"Two triangles: triangle ABC with sides AB=6 and AC=7, and triangle DEF with sides DF=14 and EF=x.\" width=\"455\" height=\"175\" class=\"aligncenter size-full wp-image-273649\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-2-1\">12<\/div><div class=\"PQ\"  id=\"PQ-2-2\">18<\/div><div class=\"PQ\"  id=\"PQ-2-3\">3<\/div><div class=\"PQ\"  id=\"PQ-2-4\">6<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Because the triangles are similar, their sides are proportional. This means that \\(\\overline{BC}\\) is proportional to \\(\\overline{EF}\\), and \\(\\overline{AC}\\) is proportional to \\(\\overline{DF}\\).<\/p>\n<p>Set up a proportion.<\/p>\n<p style=\"text-align:center;\">\\(\\dfrac{6}{x}=\\dfrac{7}{14}\\)<\/p>\n<p>Cross multiply and solve for \\(x\\).<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(6\\times14=7x\\)<br \/>\n\\(84=7x\\)<br \/>\n\\(12=x\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\n\\(\\triangle\\)ABC is similar to \\(\\triangle\\)DEF. Find \\(x\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Similar-triangles-example-4.svg\" alt=\"Two triangles are shown. Triangle ABC has angles 46\u00b0 at B and 78\u00b0 at A. Triangle DEF has angles x at E and 78\u00b0 at D.\" width=\"320\" height=\"176\" class=\"aligncenter size-full wp-image-273664\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">81\u00b0<\/div><div class=\"PQ\"  id=\"PQ-3-2\">31\u00b0<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-3\">46\u00b0<\/div><div class=\"PQ\"  id=\"PQ-3-4\">56\u00b0<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Similar triangles have congruent angle measures. Since \\(\\angle B = 46\u00b0\\), \u2220E also must be 46\u00b0.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nAre these two triangles similar?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Similar-triangles-example-5.svg\" alt=\"Two triangles, one smaller and one larger, each with angles labeled 93\u00b0 and 47\u00b0 in the same positions.\" width=\"427\" height=\"110\" class=\"aligncenter size-full wp-image-273667\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-4-1\">Yes<\/div><div class=\"PQ\"  id=\"PQ-4-2\">No<\/div><div class=\"PQ\"  id=\"PQ-4-3\">Sometimes<\/div><div class=\"PQ\"  id=\"PQ-4-4\">Cannot be determined by given information<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Similar triangles must have congruent angle measures. Since two of the angles are known in each triangle and are the same, they can be determined to be similar triangles because the third angles must also be the same.<\/p>\n<p>All triangles have interior angles that add to 180\u00b0, so the third angle in both these triangles must be as follows:<\/p>\n<p style=\"text-align:center;\">\n\\(180\u00b0-93\u00b0-47\u00b0=30\u00b0\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\n\\(\\triangle\\)ABC is similar to \\(\\triangle\\)DEF. Find \\(x\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2025\/10\/Similar-triangles-example-6.svg\" alt=\"Two triangles are shown: triangle ABC with sides AB = 13 and BC = 21, and triangle DEF with sides DE = 27 and EF = x.\" width=\"457\" height=\"169\" class=\"aligncenter size-full wp-image-273670\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">58.53<\/div><div class=\"PQ\"  id=\"PQ-5-2\">52.27<\/div><div class=\"PQ\"  id=\"PQ-5-3\">47.19<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-4\">43.62<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Because the triangles are similar, their sides are proportional. This means that \\(\\overline{AB}\\) is proportional to \\(\\overline{DE}\\) and  \\(\\overline{BC}\\) is proportional to \\(\\overline{EF}\\).<\/p>\n<p>Set up a proportion.<\/p>\n<p style=\"text-align:center;\">\n\\(\\dfrac{13}{27}=\\dfrac{21}{x}\\)<\/p>\n<p>Cross multiply and solve for \\(x\\).<\/p>\n<p style=\"text-align: center; line-height: 35px\">\n\\(13x=21\\times27\\)<br \/>\n\\(13x=567\\)<br \/>\n\\(x \\approx 43.62\\)<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/geometry\/\">Return to Geometry Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Geometry Videos<\/p>\n","protected":false},"author":1,"featured_media":100237,"parent":0,"menu_order":25,"comment_status":"closed","ping_status":"open","template":"","meta":{"footnotes":""},"class_list":{"0":"post-666","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-triangle-videos","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/666","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=666"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/666\/revisions"}],"predecessor-version":[{"id":212656,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/666\/revisions\/212656"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100237"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=666"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}