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Transcript
Geometry, You Can Do It ! Similar polygons When two polygons have the same shape and differ only in size, we say they are similar polygons. by Bill Hanlon Angle, Angle Postulate If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar Using that postulate, we can move on and to make some more observations These two squares are similar. More formally, two polygons are similar if and only if there is a one to one correspondence between their vertices such that: Theorem 1. Corresponding angles are congruent. If an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides including those angles are proportional, the triangles are similar. 2. Lengths of corresponding sides are in proportion. Let’s see if we can prove two triangles are similar with the information we have learned. A Isn’t that unbelievable ? We just get done having a lesson on proportion and here we are using it in a definition.Don’t you just love math ? D X So similar polygons look the same; the angles are congruent and their sides are in proportion, respectively. C B Given: Prove: Using a property of proportion, we could easily prove the following theorem. AC ll BD ∆ AXC ~ ∆ BXD Statements Reasons Theorem If two polygons are similar, the ratio of their perimeters equals the ratio of the lengths of any pair of corresponding sides. A’ A B C B’ perimeter ∆ABC perimeter ∆A’B’C’ C’ A A’B’ Up to this point, the only way we could show polygons are similar is by using the definition and meeting those conditions. If we were to play with triangles that had the same shapes, we would eventually notice something. We wil state our observation in terms of a postulate. Tape 10 1. AC ll BD Given 2. ∠A ≅ ∠B 2 ll lines cut by t, alt int ∠ ‘s are congr 3. ∠AXC ≅ ∠BXD Vert ∠ ‘s ≅ 4. ∆AXC ~ BXD 2 ∠ ‘s of ∆ ≅ to 2 ∠ ‘s of another ∆ Once we know that triangles are similar, we then know, by definition, that the sides are in proportion. That will allow us to determine lengths of triangles. www.hanlonmath.com ©1997 Mathematical Systems