Angles, Degrees, and Special Triangles

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Angles, Degrees, and Special Triangles

• Trigonometry
• MATH 103
• S. Rook

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Overview

• Section 1.1 in the textbook
• Angles
• Degree measure
• Triangles
• Special Triangles

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Angles
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Angles

• Angle describes the space between two rays
  that are joined at a common endpoint
• Recall from Geometry that a ray has one
  terminating side and one non-terminating side
• Can also think about an angle as a rotation about
  the common endpoint
• Start at OA (Initial side)
• End at OB (Terminal side)

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Angles (Continued)

• If the initial side is rotated
• counter-clockwise
• ? is a positive angle
• If the initial side is rotated
• clockwise
• ? is a negative angle

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Degree Measure
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Degree Measure

• Degree measure expresses the size of an angle.
  Often abbreviated by the symbol
• 360 makes one complete revolution
• The initial and terminal sides of the angle are
  the same
• 180 makes one half of a complete revolution
• 90 makes one quarter of a complete revolution

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Degree Measure (Continued)

• Angles that measure
• Between 0 and 90 are known as acute angles
• Exactly 90 are known as right angles
• Denoted by a small square between the initial and
  terminal sides
• Between 90 and 180 are known as obtuse angles
• Complementary angles two angles whose measures
  sum to 90
• Supplementary angles two angles whose measures
  sum to 180

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Degree Measure (Example)

• Ex 1 (i) Indicate whether the angle is acute,
  right, or obtuse (ii) find its complement (iii)
  find its supplement
• a) 50
• b) 160

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Triangles
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Triangles

• Triangle a polygon comprised of three sides and
  three angles the sum of which add to 180
• The longest side is opposite the largest angle
  measure and the smallest side is opposite the
  smallest angle measure
• Important types of triangles
• Equilateral all three sides are of equal length
  and all three angles are of equal measure
• Isosceles two of the sides are of equal length
  and two of the angles are of equal measure
• Scalene all sides have a different length and
  all angles have a different measure

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Triangles (Continued)

• Triangles can also be classified based on the
  measurement of their angles
• Acute triangle all angles of the triangle are
  acute
• Obtuse triangle one angle of the triangle is
  obtuse
• Right triangle one angle of the triangle is a
  right angle
• VERY important

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Special Triangles Right Triangle

• Pythagorean Theorem a2 b2 c2 where a and b
  are the legs of the triangle and c is the
  hypotenuse
• The legs are the shorter sides of the triangle
• The hypotenuse is the longest side of the
  triangle and is opposite the 90 angle
• Can be used when we have information regarding at
  least two sides of the triangle
• The Pythagorean Theorem can ONLY be used with a
  RIGHT triangle

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Special Triangles Right Triangle (Example)

• Ex 2 Find the length of the missing side
• a)
• b) If a 2 and c 6, find b

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Special Triangles 30 - 60 - 90 Triangle

• Think about taking half of an equilateral
  triangle
• Shortest side is x and is opposite the 30 angle
• Medium side is and is opposite the 60
  angle
• Longest side is 2x and is
• opposite the 90 angle

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Special Triangles 30 - 60 - 90 Triangle
(Example)

• Ex 3 Find the length of the remaining sides
• a)
• b) The side opposite 60 is 4

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Special Triangles 45 - 45 - 90

• Think about taking half of a square along its
  diagonal
• Shortest sides are x and are opposite the 45
  angles
• Longest side is and is
• opposite the 90 angle

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Special Triangles 45 - 45 - 90 Triangle
(Example)

• Ex 4 Find the length of the remaining sides
• a)
• b) The longest side is

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Summary

• After studying these slides, you should be able
  to
• Understand angles and angle measurement
• Identify the complement or supplement of an angle
• Find the third side of a right triangle when
  given two sides
• Find the length of any side of a 30-60-90
  triangle given the length of one of its sides
• Find the length of any side of a 45-45-90
  triangle given the length of one of its sides
• Additional Practice
• See the list of suggested problems for 1.1
• Next lesson
• The Rectangular Coordinate System (Section 1.2)