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Understanding Triangle Properties: Angles, Similarity, and the Pythagorean Theorem

This chapter explores essential concepts in triangle geometry. Learn to solve problems involving angle relationships in triangles and derive the measures of unknown angles using Euclid’s Theorem, which states that the sum of the angles in any triangle is 180º. Understand similarity in triangles through corresponding angles and sides, and apply the Pythagorean Theorem to calculate unknown lengths in right triangles. This comprehensive guide provides examples and proofs to solidify your understanding of triangle geometry.

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Understanding Triangle Properties: Angles, Similarity, and the Pythagorean Theorem

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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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