2.1 Lines and Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle 

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

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

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 & 

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 & 

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

2.2 Triangles Triangles classified by number of congruent sides Types of triangles# sides congruent scalene0 isosceles2 equilateral3

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

2.2 Triangles In a triangle, the sum of the interior angle measures is 180º (m  A + m  B + m  C = 180º) A B C

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 

2.2 Triangles Perimeter of triangle = sum of lengths of sides Area of a triangle = ½ base  height h b

2.2 Triangles Heron’s 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 Heron’s formula instead of A = ½ bh?

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”

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

2.3 Quadrilaterals Quadrilateral ParallelogramTrapezoid Rectangle Rhombus Square Isosceles Trapezoid

2.3 Quadrilaterals PolygonArea Squares2s2 Rectangle l  w Parallelogram b  h Triangle Trapezoid

2.3 Quadrilaterals PolygonPerimeter Trianglea + b + c (3 sides) Quadrilaterala + b + c + d (4 sides) Parallelogram2a + 2b Rectangle2l + 2w Square4s

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

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

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?

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)?

3.2 More About Functions Graph of What is the domain and the range?

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

4.1 Angles Acute angle – 0  < x < 90  Right angle - 90  Obtuse angle – 90  < x < 180  Straight angle 

4.1 Angles 45  angle Also = 315  135  angle Also = 225 

4.1 Angles Converting degrees to radians (definition): Examples:

4.1 Angles Standard position – always w.r.t. x-axis

4.2 Defining the Trigonometric Functions Diagram: y r x

4.2 Defining the Trigonometric Functions Definitions: y r x

4.2 Defining the Trigonometric Functions Given one function – find others : y r x

4.3 Values of the trigonometric functions triangle: –Leg opposite the 45  angle = a –Leg opposite the 90  angle = a a 45  90  45 

4.3 Values of the trigonometric functions triangle: –Leg opposite 30  angle = a –Leg opposite 60  angle = –Leg opposite 90  angle = 2a a 2a 90  30  60 

4.3 Values of the trigonometric functions Common angles for trigonometry

4.3 Values of the trigonometric functions Some common trig function values:

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:

4.3 Values of the trigonometric functions Some common inverse trig function values:

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

4.4 The Right Triangle Given the right triangle oriented as follows: a c b C B A

4.4 The Right Triangle Example: Given A = 30 , a = 2, solve the triangle. a c b C B A

4.4 The Right Triangle Example: Solve the triangle given: a c b C B A

4.5 Applications of Right Triangles No new material – applications of the previous section. a c b C B A