1. 10.2 Triangles

2. 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.

3. 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°. A C B

4. 2-Column Proof Given: ΔABC Prove:  A +  ABC +  C = 180° StatementsReason 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 =  C3. If 2 || lines are cut by a transversal, alt. int. angles are equal. 4.  A +  ABC +  C = 180°4. Substitution of equals A C B 1 2 3

5. Exterior Angle Theorem Theorem: An exterior angle of a triangle equals the sum of the non-adjacent interior angles. 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 A C B D

6. Base ∠ s of Isosceles △ Theorem: Base ∠ s of an isosceles △ are equal. Given: Isosceles △ ABC, with AC = BC Prove: ∠ A = ∠ B A C B

7. Another Proof Theorem: An  formed by 2 radii subtending a chord is 2 x an inscribed  subtending the same chord. Given: Circle C, with central ∠ C and inscribed ∠ D Prove: ∠ ACB = 2 ∠ ADB C F D B A x y a d b c

8. 2-Column Proof Given: Circle C, with central ∠ C and inscribed ∠ D Prove: ∠ ACB = 2 ∠ ADB C F D B A x y a d b c StatementsReasons 1. Circle C, with central ∠ C, inscribed ∠ D 1.Given 2. Draw line CD, intersecting circle at F 2. Two points determine a 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

9. 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

10. Triangles and Their Characteristics

11. Similar Triangles △ ABC ~ △ XYZ iff Corresponding angles are equal Corresponding sides are proportional ∠ 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. A B C X Z Y

12. Example A E D C B 25 8 12 x

13. Example How can you estimate the height of a building when you know your own height (on a sunny day). 400 6 10 x 6 10 --- = ------ x 400 6 400 = 10 x x = 240

14. Your Turn

15. 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 a 2 + b 2 = c 2

16. Example C

17. Your Turn A C B 11 8 b Γ