Lesson Similarity tests for right-angled triangles
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<H2>Similarity tests for right-angled triangles</H2> For right-angled triangles the similarity tests are read as follows: 1. If two right-angled triangles have one acute angle congruent then the triangles are similar. This is a consequence of a general <B>AA</B> similarity test for arbitrary triangles (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Triangles/Similarity-tests-for-triangles.lesson>Similarity tests for triangles</A> under the current topic in this site). Indeed, if two right-angled triangles have one pair of acute angles congruent then the other pair of the acute angles are congruent too as the <B>complementary</B> angles (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). 2. If the two legs of a right-angled triangle are proportional in length to the two legs of the other right-angled triangle then the triangles are similar. This is the consequence of a general <B>SAS</B> similarity test for arbitrary triangles (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Triangles/Similarity-tests-for-triangles.lesson>Similarity tests for triangles</A> under the current topic in this site) applied to right-angled triangles. 3. If the hypotenuse and one leg of a right-angled triangle are proportional in length to the hypotenuse and a leg of the other right-angled triangle then the triangles are similar. <H3>Example 1</H3>All right-angled isosceles triangles are similar. Indeed, they all have acute angles of 45°. <H3>Example 2</H3>All right-angled (30°, 60°) triangles are similar. Indeed, they all have congruent acute angles of 30°. <H3>Example 3</H3>One right-angled triangle has an acute angle of 42°. The other right-angled triangle has an acute angle of 48°. Prove that these right-angled triangles are congruent. <B>Proof</B> The second right-angled triangle has the other acute angle equal to 90° - 48° = 42°. Thus these two right-angled triangles have congruent angles of 42°. Therefore, the triangles are similar in accordance with the <B>AA</B> similarity test for right-angled triangles (see above in this lesson). 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/Similarity-tests-for-triangles.lesson>Similarity tests for 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/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>.
