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Lesson 97 Angles & Triangles Pythagorean Theorem Pythagorean Triples

LESSON PRESENTATION. Example 97.1. Example 97.2. Example 97.3. Example 97.4. Lesson 97 Angles & Triangles Pythagorean Theorem Pythagorean Triples. Angles & Triangles. When two lines intersect they form four angles. Here are shown two intersecting lines and the four angles that are formed.

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Lesson 97 Angles & Triangles Pythagorean Theorem Pythagorean Triples

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  1. LESSON PRESENTATION Example 97.1 Example 97.2 Example 97.3 Example 97.4 Lesson 97Angles & TrianglesPythagorean TheoremPythagorean Triples

  2. Angles & Triangles When two lines intersect they form four angles. Here are shown two intersecting lines and the four angles that are formed. Line 1 A B D C Line 2

  3. Angles & Triangles If we look at angle B, we see that it is formed by part of Line 1 and Line 2. We call these parts rays. Line 1 x x x x The x’s represent the exterior of the angle. The dots represent the interior of the angle. x x B x x x x Line 2 An angle is formed by two rays with a common endpoint. The rays and the common endpoint are considered the angle itself.

  4. Angles & Triangles Also, we should remember that when two straight lines intersect and are perpendicular to each other, they form four angles that we define as 90 degree angles. 90o 90o 90o 90o

  5. Angles & Triangles 60o 25o 60o 135o 30o 60o 60o 20o Using this definition of a 90o angle and two axioms, it can be proved, using geometry, that the sum of the interior angles of any triangle is 180o. Below there are three triangles. Note that the sum of the interior angles in each triangle is 180o. 90 + 60 + 30 = 180 60 + 60 + 60 = 180 135 + 25 + 20 = 180

  6. Angles & Triangles The triangle below has one angle that has a measure of 90o. Any triangle that contains a right angle is called a right triangle and the side that is opposite the right angle is always the longest side and is called the hypotenuse. The other two sides are called legs, or simply, sides. Hypotenuse Side or Leg Side or Leg

  7. Pythagorean Theorem 5 4 3 It can be shown that the square drawn on the hypotenuse of a right triangle has the same area as the sum of the areas of the squares drawn on the other two sides.

  8. Pythagorean Theorem c a b Again: The square of the hypotenuse is equal to the sum of the squares of the other two sides.

  9. Example 97.1 5 4 a Given the triangle with the lengths of the sides as shown, use the Pythagorean Theorem to find a. The length of side a is 3 units. Although –3 and +3 are both solutions of the equation a2 = 9, the only solution that makes sense here is a = 3 because physical lengths are always positive.

  10. Example 97.2 p 5 4 Find side p. Since sides of triangles do not have negative lengths, we discard the negative result and say

  11. Example 97.3 k 5 Find side k. Since –6 has no meaning as the length of the side of a triangle, we say

  12. Example 97.4 Find side m. 8 12 m Note that when solving for m, we do not consider negative values for m since it represents a length.

  13. Pythagorean Triples 17 13 8 5 5 3 4 12 15 It is useful to commit to memory the lengths of the sides of certain right triangles. We show some of these right triangles below. 5-12-13 Right Triangle 8-15-17 Right Triangle 3-4-5 Right Triangle

  14. Pythagorean Triples 15 9 10 6 5 3 4 8 12 Not only do the Pythagorean triples just mentioned work. Pythagorean triples can also be multiples of the ones just mentioned. 6-8-10 Right Triangle 9-12-15 Right Triangle 3-4-5 Right Triangle

  15. Example 97.5 17 a 8 5 b 12 c 8 10 Pythagorean Triples Recall the appropriate Pythagorean triples to find the unknown length in each of the following right triangles. (13) (15) (6)

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