1. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-1 Chapter 1 Section 9-4 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem

2. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-2 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem Congruent Triangles Similar Triangles The Pythagorean Theorem

3. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-3 Congruent Triangles A B C D E F

4. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-4 Congruence Properties - SAS 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.

5. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-5 Congruence Properties - ASA 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.

6. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-6 Congruence Properties - SSS 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.

7. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-7 Example: Proving Congruence (SAS) Given:CE = ED AE = EB Prove: A B D E C Proof

8. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-8 Example: Proving Congruence (ASA) Given: Prove: A B D C Proof

9. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-9 Example: Proving Congruence (SSS) Given:AD = CD AB = CB Prove: A B D C Proof

10. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-10 Important Statements About Isosceles Triangles If ∆ABC is an isosceles triangle with AB = CB, and if D is the midpoint of the base AC, then the following properties hold. A C B D

11. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-11 Similar Triangles Similar Triangles are pairs of triangles that are exactly the same shape, but not necessarily the same size. The following conditions must hold.

12. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-12 Angle-Angle (AA) Similarity Property 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.

13. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-13 Example: Finding Side Length in Similar Triangles Find the length of side DF. A B C D E F Solution 16 24 32 8

14. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-14 Pythagorean Theorem If the two legs of a right triangle have lengths a and b, and the hypotenuse has length c, then That is, the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.

15. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-15 Example: Using the Pythagorean Theorem Find the length a in the right triangle below. Solution 39 36 a

16. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-16 Converse of the Pythagorean Theorem If the sides of lengths a, b, and c, where c is the length of the longest side, and if then the triangle is a right triangle.

17. © 2008 Pearson Addison-Wesley. All rights reserved 9-4-17 Example: Applying the Converse of the Pythagorean Theorem Is a triangle with sides of length 4, 7, and 8, a right triangle? Solution.