2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90 ...

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Transcript of 2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90 ...

  • Slide 1
  • 2.1 Lines and Angles Acute angle 0 < x < 90 Right angle - 90 Obtuse angle 90 < x < 180 Straight angle - 180
  • Slide 2
  • 2.1 Lines and Angles Complementary angles add up to 90 Supplementary angles add up to 180 Vertical angles the angles opposite each other are congruent
  • Slide 3
  • 2.1 Lines and Angles Intersection 2 lines intersect if they have one point in common. Perpendicular 2 lines are perpendicular if they intersect and form right angles Parallel 2 lines are parallel if they are in the same plane but do not intersect
  • Slide 4
  • 2.1 Lines and Angles When 2 parallel lines are cut by a transversal the following congruent pairs of angles are formed: Corresponding angles: 1 & 5, 2 & 6, 3 & 7, 4 & 8 Alternate interior angles: 4 & 5, 3 & 6 Alternate exterior angles: 1 & 8, 2 & 7 12 34 56 78
  • Slide 5
  • 2.1 Lines and Angles When 2 parallel lines are cut by a transversal the following supplementary pairs of angles are formed: Same side interior angles: 3 & 5, 4 & 6 Same side exterior angles: 1 & 7, 2 & 8 12 34 56 78
  • Slide 6
  • 2.1 Lines and Angles When 3 or more parallel lines are cut by a pair of transversals, the transversals are divided proportionally by the parallel lines AD EB CF
  • Slide 7
  • 2.2 Triangles Triangles classified by number of congruent sides Types of triangles# sides congruent scalene0 isosceles2 equilateral3
  • Slide 8
  • 2.4 The Angles of a Triangle Triangles classified by angles Types of trianglesAngles acuteAll angles acute obtuseOne obtuse angle rightOne right angle equiangularAll angles congruent
  • Slide 9
  • 2.2 Triangles In a triangle, the sum of the interior angle measures is 180 (m A + m B + m C = 180) A B C
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  • 2.2 Triangles The measure of an exterior angle of a triangle equals the sum of the measures of the 2 non- adjacent interior angles - m 1 + m 2 = m 4 1 2 34
  • Slide 11
  • 2.2 Triangles Perimeter of triangle = sum of lengths of sides Area of a triangle = base height h b
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  • 2.2 Triangles Herons formula If 3 sides of a triangle have lengths a, b, and c, then the area A of a triangle is given by: Why use Herons formula instead of A = bh?
  • Slide 13
  • 2.2 Triangles Definition: Two Triangles are similar two conditions are satisfied: 1.All corresponding pairs of angles are congruent. 2.All corresponding pairs of sides are proportional. Note: ~ is read is similar to
  • Slide 14
  • 2.2 Triangles Given ABC ~ DEF with the following measures, find the lengths DF and EF: A C 6 D F E 5 4 B 10
  • Slide 15
  • 2.3 Quadrilaterals Quadrilateral ParallelogramTrapezoid Rectangle Rhombus Square Isosceles Trapezoid
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  • 2.3 Quadrilaterals PolygonArea Squares2s2 Rectangle l w Parallelogram b h Triangle Trapezoid
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  • 2.3 Quadrilaterals PolygonPerimeter Trianglea + b + c (3 sides) Quadrilaterala + b + c + d (4 sides) Parallelogram2a + 2b Rectangle2l + 2w Square4s
  • Slide 18
  • 2.4 Circumference and Area of a Circle Circumference of a circle: C = d = 2 r 22/7 or 3.14 Area of a circle Note: Just need area and circumference formulas from this section r
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  • 2.6 Solid Geometric Figures V = volumeA = total surface area S = lateral surface area Rectangular solidV=lwhA=2lw+2lh+2wh CubeV=e 3 A=6s 2 Right circular cylinder V= r 2 hA=2 r 2 +2 rhS=2 rh Right prismV=BhA=2B+phS=ph Right circular cone V=(1/3) r 2 hA= r 2 + rsS= rs Regular pyramidV=(1/3)BhA=B+(1/2)psS=(1/2)ps Sphere V=(4/3) r 3 A=4 r 2
  • Slide 20
  • 3.2 More About Functions Domain: x-values (input) Range: y-values (output) Example: Demand for a product depends on its price. Question: If a price could produce more than one demand would the relation be useful?
  • Slide 21
  • 3.2 More About Functions Function notation: y = f(x) read y equals f of x note: this is not f times x Linear function: f(x) = mx + b Example: f(x) = 5x + 3 What is f(2)?
  • Slide 22
  • 3.2 More About Functions Graph of What is the domain and the range?
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  • 3.2 More About Functions - Determining Whether a Relation or Graph is a Function A relation is a function if: for each x-value there is exactly one y-value Function: {(1, 1), (3, 9), (5, 25)} Not a function: {(1, 1), (1, 2), (1, 3)} Vertical Line Test if any vertical line intersects the graph in more than one point, then the graph does not represent a function
  • Slide 24
  • 4.1 Angles Acute angle 0 < x < 90 Right angle - 90 Obtuse angle 90 < x < 180 Straight angle - 180
  • Slide 25
  • 4.1 Angles 45 angle Also 360-45 = 315 135 angle Also 360-135 = 225
  • Slide 26
  • 4.1 Angles Converting degrees to radians (definition): Examples:
  • Slide 27
  • 4.1 Angles Standard position always w.r.t. x-axis
  • Slide 28
  • 4.2 Defining the Trigonometric Functions Diagram: y r x
  • Slide 29
  • 4.2 Defining the Trigonometric Functions Definitions: y r x
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  • 4.2 Defining the Trigonometric Functions Given one function find others : y r x
  • Slide 31
  • 4.3 Values of the trigonometric functions 45-45-90 triangle: Leg opposite the 45 angle = a Leg opposite the 90 angle = a a 45 90 45
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  • 4.3 Values of the trigonometric functions 30-60-90 triangle: Leg opposite 30 angle = a Leg opposite 60 angle = Leg opposite 90 angle = 2a a 2a 90 30 60
  • Slide 33
  • 4.3 Values of the trigonometric functions Common angles for trigonometry
  • Slide 34
  • 4.3 Values of the trigonometric functions Some common trig function values:
  • Slide 35
  • 4.3 Values of the trigonometric functions The inverse trigonometric functions are defined as the angle giving the result for the given function (sin, cos, tan, etc.) Example: Note:
  • Slide 36
  • 4.3 Values of the trigonometric functions Some common inverse trig function values:
  • Slide 37
  • 4.4 The Right Triangle Solving a triangle: Given 3 parts of a triangle (at least one being a side), we are to find the other 3 parts. Solving a right triangle: Since one angle is 90 , we need to know another angle (the third angle will be the complement) and a side or we need to know 2 of 3 sides (use the Pythagorean theorem to find 3 rd side). a c b C B A
  • Slide 38
  • 4.4 The Right Triangle Given the right triangle oriented as follows: a c b C B A
  • Slide 39
  • 4.4 The Right Triangle Example: Given A = 30 , a = 2, solve the triangle. a c b C B A
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  • 4.4 The Right Triangle Example: Solve the triangle given: a c b C B A
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  • 4.5 Applications of Right Triangles No new material applications of the previous section. a c b C B A