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Section 9-3

Section 9-3. The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem. The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem. Congruent Triangles Similar Triangles The Pythagorean Theorem. Congruent Triangles.

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Section 9-3

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  1. Section 9-3 • The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem

  2. The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem • Congruent Triangles • Similar Triangles • The Pythagorean Theorem

  3. Congruent Triangles Triangles that are both the same size and same shape are called congruent triangles. B E A D C F The corresponding sides are congruent and corresponding angles have equal measures. Notation:

  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. 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. 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. Example: Proving Congruence (SAS) Given: CE = ED AE = EB Prove: C B E A D Proof

  8. Example: Proving Congruence (ASA) Given: Prove: B C A D Proof

  9. Example: Proving Congruence (SSS) Given: AD = CD AB = CB Prove: B A C Proof D

  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. 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. B A D C

  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. 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.

  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. Example: Finding Side Length in Similar Triangles E 8 B F Find the length of side DF. 16 24 D Solution C 32 Set up a proportion with corresponding sides: A Solving, we find that DF = 16.

  14. Gougo’sTheorem 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. hypotenuse c leg a leg b

  15. Example: Using the Pythagorean Theorem Find the length a in the right triangle below. 39 a Solution 36

  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. Example: Applying the Converse of the Pythagorean Theorem Is a triangle with sides of length 4, 7, and 8, a right triangle? Solution ? ? No, it is not a right triangle.

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