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Chapter 1 Chapter 9 Geometry Section 9-4 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem © 2008 Pearson Addison-Wesley. All rights reserved 9-4-2 © 2008 Pearson Addison-Wesley. All rights reserved The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem Congruent Triangles Triangles that are both the same size and same shape are called congruent triangles. B E • Congruent Triangles • Similar Triangles • The Pythagorean Theorem A D F C The corresponding sides are congruent and corresponding angles have equal measures. Notation: ∆ABC ≅ ∆DEF . 9-4-3 © 2008 Pearson Addison-Wesley. All rights reserved 9-4-4 © 2008 Pearson Addison-Wesley. All rights reserved Congruence Properties - SAS Congruence Properties - ASA Side-Angle-Side (SAS) If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent. Angle-Side-Angle (ASA) If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent. 9-4-5 © 2008 Pearson Addison-Wesley. All rights reserved 9-4-6 © 2008 Pearson Addison-Wesley. All rights reserved 1 Congruence Properties - SSS Example: Proving Congruence (SAS) Given: Side-Side-Side (SSS) If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent. CE = ED AE = EB ∆ACE ≅ ∆BDE Prove: Proof STATEMENTS C B E A D REASONS 1. CE = ED 1. Given 2. AE = EB 2. Given 3. ∠CEA = ∠BED 3. Vertical Angles are equal 4. ∆ACE ≅ ∆BDE 4. SAS property 9-4-7 9-4-8 © 2008 Pearson Addison-Wesley. All rights reserved © 2008 Pearson Addison-Wesley. All rights reserved Example: Proving Congruence (ASA) Given: ∠ADB = ∠DBC ∠ABD = ∠CDB ∆ADB ≅ ∆CDB Prove: Proof STATEMENTS B Given: C A Example: Proving Congruence (SSS) AD = CD AB = CB ∆ABD ≅ ∆CDB Prove: Proof STATEMENTS D REASONS B A REASONS D 1. ∠ADB = ∠DBC 1. Given 2. ∠ABD = ∠CDB 2. Given 1. AD = CD 1. Given 2. AB = CB 2. Given 3. DB = DB 3. Reflexive property 3. BD = BD 3. Reflexive property 4. ∆ADB ≅ ∆CDB 4. ASA property 4. ∆ABD ≅ ∆CDB 4. SSS property 9-4-9 © 2008 Pearson Addison-Wesley. All rights reserved 9-4-10 © 2008 Pearson Addison-Wesley. All rights reserved Important Facts About Isosceles Triangles Similar Triangles If ∆ABC is an isosceles triangle with AB = CB, and if D is the midpoint of the base AC, then the B following properties hold. Similar Triangles are pairs of triangles that are exactly the same shape, but not necessarily the same size. The following conditions must hold. 1. The base angles A and C are equal. 2. Angles ABD and CBD are equal. 3. Angles ADB and CDB are both right angles. 1. Corresponding angles must have the same measure. 2. The ratios of the corresponding sides must be constant; that is, the corresponding sides are proportional. A D C 9-4-11 © 2008 Pearson Addison-Wesley. All rights reserved C 9-4-12 © 2008 Pearson Addison-Wesley. All rights reserved 2 Angle-Angle (AA) Similarity Property Example: Finding Side Length in Similar Triangles E ∆ABC is similar to ∆DEF . Find the length of side DF. If the measures of two angles of one triangle are equal to those of two corresponding angles of a second triangle, then the two triangles are similar. F B 16 24 D Solution A Solving, we find that DF = 16. 9-4-13 9-4-14 © 2008 Pearson Addison-Wesley. All rights reserved © 2008 Pearson Addison-Wesley. All rights reserved Example: Using the Pythagorean Theorem Pythagorean Theorem Find the length a in the right triangle below. If the two legs of a right triangle have lengths a and b, and the hypotenuse has length c, then a 2 + b2 = c2 . a 2 + b2 = c 2 36 a 2 + 362 = 392 a 2 + 1296 = 1521 a 2 = 225 a = 15 hypotenuse c leg b 39 a Solution That is, the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse. leg a C 32 Set up a proportion with corresponding sides: EF DF = BC AC 8 DF = 16 32 8 9-4-15 9-4-16 © 2008 Pearson Addison-Wesley. All rights reserved © 2008 Pearson Addison-Wesley. All rights reserved Example: Applying the Converse of the Pythagorean Theorem Converse of the Pythagorean Theorem Is a triangle with sides of length 4, 7, and 8, a right triangle? If the sides of lengths a, b, and c, where c is the length of the longest side, and if a 2 + b2 = c2 , then the triangle is a right triangle. Solution ? 4 2 + 7 2 = 82 ? 16 + 49 = 64 65 ≠ 64 No, it is not a right triangle. 9-4-17 © 2008 Pearson Addison-Wesley. All rights reserved 9-4-18 © 2008 Pearson Addison-Wesley. All rights reserved 3