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10.2 Triangles Axioms and Theorems • Postulate—A statement accepted as true without proof. • E.g. Given a line and a point not on the line, one and only one can be drawn through the point parallel to the given line. Theorem • Theorem—A statement that is proved from postulates, axioms, and other theorems. • E.g. The sum of the measures of the three angles of any triangle is 180°. B A C 2-Column Proof B • Given: ΔABC Prove: A + ABC + C = 180° 1 2 3 A C Statements Reason 1. Draw a line through B parallel to AC. 1. Given a line and a point not on the line, one line can be drawn through the point parallel to the line 2. 1 + 2 + 3 = 180° 2. Definition of a straight angle 3. 1 = A; 3 = C 3. If 2 || lines are cut by a transversal, alt. int. angles are equal. 4. A + ABC + C = 180° 4. Substitution of equals Exterior Angle Theorem • Theorem: An exterior angle of a triangle equals the sum of the non-adjacent interior angles. C A B D • Given: △ ABC with exterior angle ∠CBD • Prove: ∠CBD = ∠A + ∠C ∠CBD + ∠ABC = 180 -- def. of supplementary ∠s ∠A + ∠C + ∠ABC = 180 -- sum of △’s interior ∠s ∠A + ∠C + ∠ABC = ∠CBD + ∠ABC -- axiom ∠A + ∠C = ∠CBD -- axiom Base ∠s of Isosceles △ • Theorem: Base ∠s of an isosceles △ are equal. C A B • Given: Isosceles△ ABC, with AC = BC • Prove: ∠A = ∠B Another Proof • Theorem: An formed by 2 radii subtending a chord is 2 x an inscribed subtending the same chord. A F d x C c y b B a D • Given: Circle C, with central ∠C and inscribed ∠D • Prove: ∠ACB = 2 • ∠ADB 2-Column Proof • Given: Circle C, with central ∠C and inscribed ∠D • Prove: ∠ACB = 2 • ∠ADB A d x C Statements 1. Circle C, with central ∠C, inscribed ∠D Reasons 1. Given D F y b c a 2. Draw line CD, intersecting 2. Two points determine a circle at F line. 3. ∠y = ∠a + ∠b = 2 ∠a ∠x = ∠c + ∠d = 2 ∠c 3. Exterior ∠; Isosceles △ angles 4. ∠x + ∠y = 2(∠a + ∠c) 4. Equal to same quantity 5. ∠ACB = 2 ∠ADB B Your Turn • Find the measures of angles 1 through 5. • Solution: 1 = 90º 2 = 180 – (43 + 90) = 180 – (133) = 47º 3 = 47º 4 = 180 – (47 + 60) = 180 – (107) = 73º 5 = 180 – 73 = 107º Triangles and Their Characteristics Similar Triangles B △ABC ~ △XYZ iff • Corresponding angles A are equal • Corresponding sides are proportional C Y • • ∠A = ∠X; ∠B = ∠Y; ∠C = ∠Z AB/XY = BC/YZ = AC/XZ • Theorem: If 2 corresponding ∠s of 2 △s are equal, then △s are similar. X Z Example A 25 x B C D 8 12 E Example • How can you estimate the height of a building when you know your own height (on a sunny day). 6 10 --- = -----x 400 6 • 400 = 10 • x x x = 240 6 400 10 Your Turn Pythagorean Theorem • The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse. • If triangle ABC is a right triangle with hypotenuse c, then a2 + b2 = c2 Example C Your Turn B 11 8 Γ C A b