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Transcript
Geometry - Chapter 18 Similar Triangles Key Concepts This lesson is devoted to similar triangles. We are going to learn lots of new theorems and applications on how to solve and identify similar triangles. This lesson is a lot like the lesson where we proved that triangles are congruent. It relates also to our last lesson, so we should breeze right through it! Section 18-1 Like congruent triangles, there are lots of theorems and corollaries to help us identify similar triangles. That is what section 18-1 is all about. Understanding the theorems is a crucial part of this section. Let’s try a few problems to help us better understand the properties of similar triangles. Theorem 18-1 AAA Similarity Theorem If the angles of one triangle are congruent to the angles of another triangle, then the triangles are similar. Corollary 18-2 AA Similarity Corollary If two angles of one triangle are congruent to two angles of a second triangle, then the triangles are similar. Corollary 18-3 Right-Triangle Similarity Corollary If two right triangles have an acute angle of one congruent to an acute angle of the other, then the triangles are similar. Corollary 18-4 Parallel Line Similarity Corollary If a line parallel to one side of a triangle determines a second triangle, then the second triangle will be similar to the original triangle. 1 Problem: Q Given m SQ = 12 and TU SR Find TS. Solution: ∆QTU and ∆QSR are similar triangles. Therefore, all of their corresponding sides have a ratio of 1 to 4 since we know that two of their corresponding sides (TU and SR) have this ratio (2 to 8 reduces to 1 to 4). So, if SQ = 12, then TQ = 3. Then TS = 12 – 3 = 9. T 2 S U R 8 Problem: Given: Prove: PL ML = PO MN L P ∆LMP is similar to ∆LNO O Solution: We need to set up a two-column proof. Statements 1. M PL LM = PO MN N Reasons 1. Given 2. PM ON 2. By Corollary 17-4 which says, if a line divides two sides of a triangle proportionally then the line is parallel to the third side. 3. ∆LMP is similar to ∆LNO 3. By Corollary 18-4 which says, if a line is parallel to one side of a triangle determines a second triangle, then the second triangle will be similar to the original triangle. 2 Section 18-2 In this section, we are going to study similar triangles a little more deeply. Like discovering whether two triangles are congruent, there are theorems that help us to discover if two triangles are similar. It is important to look over the theorems and corollaries first, then try the practice problems before you get started on your submission problems. Theorem 18-5 SAS Similarity Theorem If an angle of one triangle is congruent to an angle of another, and the sides including these angles are proportional, then the triangles are similar. Theorem 18-6 SSS Similarity Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. Corollary 18-7 Proportional Perimeters Corollary The perimeters of two similar triangles are proportional to any pair of corresponding sides. Corollary 18-8 Proportional Altitudes Corollary The altitudes of similar triangles are proportional to any pair of corresponding sides. Corollary 18-9 Proportional Medians Corollary The medians of similar triangles are proportional to any pair of corresponding sides. Problem: If m∠A = 35 Find m∠1, m∠2& m∠3 , B 3 D 2 1 Solution: C A m∠3 = 180 − 90 − 35 = 55 m∠2 = 180 − 90 − 55 = 35 m∠1 = 90 − 35 = 55 3 That is the end of the first semester. It was kind of fun, wasn’t it? Now it’s time to get ready for your semester exam. You should review all of your submissions to be sure you understand how to do ALL the problems. Ask for help if you don’t understand! Review all your theorems, corollaries, postulates, and definitions. If you’ve been keeping up with your lessons and submissions all semester, you won’t have any difficulty with this exam. Good luck! 4