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CHAPTER 10

CHAPTER 10. Geometry. 10.2. Triangles. Objectives Solve problems involving angle relationships in triangles. Solve problems involving similar triangles. Solve problems using the Pythagorean Theorem. 3. Triangle.

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CHAPTER 10

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  1. CHAPTER 10 Geometry

  2. 10.2 Triangles

  3. Objectives Solve problems involving angle relationships in triangles. Solve problems involving similar triangles. Solve problems using the Pythagorean Theorem. 3

  4. Triangle • A closed geometric figure that has three sides, all of which lie on a flat surface or plane. • Closed geometric figures • If you start at any point and trace along the sides, you end up at the starting point. 4

  5. Euclid’s Theorem Theorem: A conclusion that is proved to be true through deductive reasoning. Euclid’s assumption: Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line. 5

  6. Euclid’s Theorem (cont.) Euclid’s Theorem: The sum of the measures of the three angles of any triangle is 180º. 6

  7. Euclid’s Theorem (cont.) Proof: m1 = m2 and m3 = m4 (alternate interior angles) Angles 2, 5, and 4 form a straight angle (180º) Therefore, m1 + m5 + m3 = 180º . 7

  8. Two-Column Proof A Theorem: The sum of ∠’s in a △ equals 180°. Given: △ABCProve: ∠A + ∠ B + ∠ACB = 180° Reasons Parallel postulate Definition of straight angle If 2 || lines, alt. int. ∠’s are equal Substitution of equals axiom C m 1 3 2 Statements • Through C, draw line m || to line AB. • ∠1 + ∠2 + ∠3 = 180° • ∠1 = ∠A; ∠3 = ∠B • ∠A + ∠ACB + ∠B = 180° A B

  9. Exterior ∠Theorem Theorem: An exterior angle of a triangle equals the sum of the non-adjacent interior angles. ∠CBD = ∠A + ∠C Since∠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 C D A B

  10. Base ∠s of Isosceles △ • Theorem: Base ∠s of an isosceles △ are equal • Given: Isosceles△ ABC, which AC = BCProve: ∠A = ∠B C B A

  11. Two-Column Proof B Theorem: An ∠ formed by 2 radii subtending a chord is 2 x an inscribed ∠ subtending the same chord. Given: Circle C with A, B, D on the circleProve: ∠ACB = 2 • ∠ADB Reasons Given 2 points determine a line (postulate) Exterior ∠; Isosceles △ angles Equal to same quantity (axiom) Substitution (axiom) A F d Statements • Circle C, with central ∠C, inscribed ∠D • Draw line CD, intersecting circle at F • ∠y = ∠a + ∠b = 2 ∠a∠x = ∠c + ∠d = 2 ∠c • ∠x + ∠y = 2(∠a + ∠c) • ∠ACB = 2 ∠ADB x y C B b c a D

  12. Example 1: Using Angle Relationships in Triangles Find the measure of angle A for the triangle ABC. Solution: mA + mB + mC =180º mA + 120º + 17º = 180º mA + 137º = 180º mA = 180º − 137º = 43º 12

  13. Example 2: Using Angle Relationships in Triangles Find the measures of angles 1 through 5. Solution: m1+ m2 + 43º =180º 90º + m2 + 43º = 180º m2 + 133º = 180º m2 = 47º 13

  14. Example 2 continued m3 + m4+ 60º =180º 47º + m4 + 60º = 180º m4 = 180º − 107º = 73º m4 + m5 = 180º 73º + m5 = 180° m5 = 107º 14

  15. Triangles and Their Characteristics 15

  16. Similar Triangles B • △ABC ~ △XYZ iff • Corresponding angles are equal • Corresponding sides are proportional • ∠A = ∠X; ∠B = ∠Y; ∠C = ∠Z • AB/XY = BC/YZ = AC/XZ • If 2 corresponding ∠s of 2 △s are equal, then △s are similar. A C Y X Z

  17. Example 1 • Given: • AB || DE • AB = 25; CE = 8 • Find AC • Solution • x 25--- = ----- 8 12 • 8(25) 25 50x = ------- = 2 • ---- = ----- ≈ 16.66 12 3 3 A 25 x B C D 8 12 E

  18. Example 2 • How can you estimate the height of a building, if you know your own height (on a sunny day)?

  19. Example 2 6 10--- = ------ x 400 6 • 400 = 10 • x x = 240 x 6 10 400

  20. Similar Triangles • Similar figures have the same shape, but not necessarily the same size. • In similar triangles, the angles are equal but the sides may or may not be the same length. • Corresponding angles are angles that have the same measure in the two triangles. • Corresponding sides are the sides opposite the corresponding angles. 20

  21. Similar Triangles (continued) Triangles ABC and DEF are similar: Corresponding AnglesCorresponding Sides Angles A and D Sides Angles C and F Sides Angles B and E Sides 21

  22. Example 3: Using Similar Triangles Find the missing length x. Solution: Because the triangles are similar, their corresponding sides are proportional: 22

  23. Example 3: Using Similar Triangles Find the missing length x. The missing length of x is 11.2 inches. 23

  24. 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 the legs have lengths a and b and the hypotenuse has length c, then a² + b² = c² 24

  25. C Example 5: Using the Pythagorean Theorem Find the length of the hypotenuse c in this right triangle: Solution: Let a = 9 and b = 12 25

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