Lesson Similarity tests for triangles
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<H2>Similarity tests for triangles</H2> In this lesson you will learn on similarity tests for triangles. The definition of similarity for triangles (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Triangles/Similar-triangles.lesson>Similar triangles</A> under the current topic in this site) includes the conditions of congruency for their corresponding angles and the conditions of proportionality for their corresponding sides. So, formally six quantities should be checked to inspect the similarity of two triangles - congruency for three pairs of the angles and proportionality for three pairs of the sides. Fortunately, the similarity tests were developed that allow to manage with lesser number of parameters. <BLOCKQUOTE><H3>AA-test on similarity for triangles</H3>If two angles of one triangle are congruent to two angles of the other triangle then the triangles are similar. <H3>SAS-test on similarity for triangles</H3>If an angle of one triangle is congruent to the angle of the other triangle and the including sides are proportional then the triangles are similar. <H3>SSS-test on similarity for triangles</H3>If three sides of one triangle are respectively proportional to the tree sides of the other triangle then the triangles are similar.</BLOCKQUOTE> Each of the three similarity tests for triangles introduced above is the set of the necessary and sufficient conditions for triangles to be similar. There are two approaches of introducing similarity tests for triangles. Under the first approach, the similarity tests are considered as the postulates and are accepted without proofs. Under another approach, the similarity tests are considered as the Theorems and are accompanied with the proofs. You may choose to which approach to follow. You can find the proofs of similarity tests in the lesson <A HREF=http://www.algebra.com/algebra/homework/Triangles/Proofs-of-Similarity-tests-for-triangles.lesson>Proofs of similarity tests for triangles</A> under the current topic in this site. Below are some typical problems on similarity tests for triangles. <H3>Problem 1</H3>If two equilateral triangles have congruent angles concluded between their lateral sides then the triangles are similar. <B>Solution</B> Since in an equilateral triangle the base angles are congruent, each of them is equal half the difference of the straight angle of 180° and the angle between the lateral sides. It implies that if two equilateral triangles have congruent angles concluded between their lateral sides then they have congruent base angles too. Therefore, such triangles are similar in accordance with the <B>AA</B> similarity test for triangles. <H3>Problem 2</H3>If an acute angle in one right-angled triangle is congruent to an acute angle in the other right-angled triangle then the triangles are similar. <B>Solution</B> The two given right-angled triangles have one pair of the corresponding congruent acute angles and the other pair of congruent angles that are the right angles. Hence, the triangles are congruent in accordance with the <B>AA</B> similarity test for triangles. By the way, these triangles have the third pair of congruent angles that are the other acute angles. They are congruent as the <B>non-adjacent complementary angles</B> (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Angles/Angles-basics.lesson>Angles basics</A> under the topic <B>Angles, complementary, supplementary angles</B> of the section <B>Geometry</B> in this site). My other lessons on similar triangles in this site are - <A HREF=http://www.algebra.com/algebra/homework/Triangles/Similar-triangles.lesson>Similar triangles</A>, - <A HREF=http://www.algebra.com/algebra/homework/Triangles/Proofs-of-Similarity-tests-for-triangles.lesson>Proofs of Similarity tests for triangles</A>, - <A HREF=http://www.algebra.com/algebra/homework/Triangles/In-a-triangle-a-straight-line-parallel-to-its-side-cuts-off-a-similar-triangle.lesson>In a triangle a straight line parallel to its side cuts off a similar triangle</A>, - <A HREF=http://www.algebra.com/algebra/homework/Triangles/Problems-on-similar-triangles.lesson>Problems on similar triangles</A>, - <A HREF=http://www.algebra.com/algebra/homework/Triangles/Similarity-tests-for-right-angled-triangles.lesson>Similarity tests for right-angled triangles</A>, - <A HREF=http://www.algebra.com/algebra/homework/Triangles/Problems-on-similarity-for-right-angled-triangles.lesson>Problems on similarity for right-angled triangles</A>, - <A HREF=http://www.algebra.com/algebra/homework/Triangles/Problems-on-similarity-for-right-angled-and-acute-triangles.lesson>Problems on similarity for right-angled and acute triangles</A>, - <A HREF=http://www.algebra.com/algebra/homework/Triangles/One-property-of-a-median-in-a-triangle.lesson>One property of a median in a triangle</A>, - <A HREF=http://www.algebra.com/algebra/homework/Triangles/One-property-of-a-trapezoid.lesson>One property of a trapezoid</A> and - <A HREF=http://www.algebra.com/algebra/homework/Triangles/Miscellaneous-problems-on-similar-triangles.lesson>Miscellaneous problems on similar triangles</A> under the current topic, and - <A HREF=http://www.algebra.com/algebra/homework/word/geometry/Solved-problems-on-similar-triangles.lesson>Solved problems on similar triangles</A> under the topic <B>Geometry</B> of the section <B>Word problems</B>. To navigate over all topics/lessons of the Online Geometry Textbook use this file/link <A HREF=https://www.algebra.com/algebra/homework/Triangles/GEOMETRY-your-online-textbook.lesson>GEOMETRY - YOUR ONLINE TEXTBOOK</A>.
