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Pythagorus Converse GEOMETRY NAME_________________________ DATE __________ Per.___________ Trigonometry 1. The Converse of the Pythagorean Theorem GIVEN: ∆ABC with CA = b, BC = a, and AB = c and a 2 + b2 = c2 . PROVE: ∆ABC is a right triangle. Let ∆XYZ be a right triangle with right angle Z and sides YZ = a and ZX = b. B a C a) Then by the Pythagorean Theorem, z2 = a 2 + b2 = c2 c b) From the GIVEN, a 2 + b2 = b 2 c) By the Transitive Property, c = A 2 2 e) Since (− z) = ( −c ) , why can’t part d)’s answer be –c = –z? Y a Z d) Taking the square root of both sides of part c gives: f) What justifies ∆XYZ ≅ ∆ABC? z b X g) Then ∠Z ≅ ∠___ because __ __ __ __ __. h) So ∠C is right and from the Definition of right triangles, 2. Pythagorean Inequalities b) Let BD = YZ, DA = ZX, and AB < XY. B Y a d a c b b D A Z Then by the Side-Side-Side Triangle Inequality, m∠D < ? 2 If c > d, then c > ___ . 2 2 2 c) If c > a + b , then ∆XYZ is: 2 2 2 a) If d = a + b , then ∆ABD is: d) Let BD = YZ, DA = ZX, but AB > XY. B Y a X d a b D A Then m∠Z < m∠D by ? Z c b 2 If c < d, then ___ < d . 2 2 2 e) If c < a + b , then ∆XYZ is: Use the SAS ∆ Inequality to complete each of the converse statements with =, <, or >. a 2 + b2 ___ c 2 . b C f) If ∆ABC is right, . f) If ∆ABC is acute, then A 2 2 2 a 2 2 2 a + b ___ c c then g) If ∆ABC is obtuse, then a + b ___ c B 3. Classify each ∆ as right, acute, or obtuse. a) a = 3, b = 4, c = 5 b) a = 4, b = 5, c = 6 c) a = 2, b = 3, c = 4 4. Find the missing integer to make a Pythagorean b) 17, 15, and ? c) 7, 25, and ? triple.a) 5, 12, and ? 5. Find x to make a right triangle. a) b) 2 x 1 x 3 1 X