Geometry: Angle Relationships

Transcription

1 Geometry: Angle Relationships I. Define the following angles (in degrees) and draw an example of each. 1. Acute 3. Right 2. Obtuse 4. Straight Complementary angles: Supplementary angles: a + b = c + d = II. Angles about a point.. make a circle! Find angle a. Vertical angles are equal Adjacent angles are supplementary!

2 Geometry: Parallel Lines McDonalds & WaWa! l m h Example: Lines l and m are parallel. Line p is perpendicular to line l. If angle y measures 40, what is the measure of angle x? 2. x h l m 1. y p 3. 2

3 Geometry: Triangles Write down formulas! I. All angles add to 180. II. Exterior angles are supplementary to interiors & add to 360. III. Special triangles. Draw the following: Isosceles Equilateral All angles = IV. Area They give you the formula. Find A, but don t be fooled by the height! Find A if b = 9 and h = 4. 3

4 Geometry: Triangles Practice

5 Geometry: Right Triangles : -The side opposite the right angle -Always the longest side Practice makes perfect! The height of a right circular cylinder is 5 and Identify Draw a picture! the diameter of its base is 4. What is the Organize the information given distance from the center of one base to a point Solve on the circumference of the other base? (A) 3 (B) 5 (C) 29 (approximately 5.39) (D) 33 (approximately 5.74) (E) 41 (approximately 6.40) 5

6 Geometry: Special Right Triangles (See Special Right Practice Problems Worksheet) I. Pythagorean Triples: (3-4-5) ( ) or multiples thereof = = II. Angle Relationships: **They give you the ratios on the test, you only need to identify the problems** ( ) ( ) When: Square Cube 45 Isosceles Right When: 30 or 60 ½ of an equilateral x = y = x = y =

7 Geometry: Polygons I. The sum of interior angles S = (n-2) x 180, Where n = the # of sides If you forget the formula, find! S = (n -2) 180 = (5-2) 180 = = 540 II. The sum of exterior angles = 360 Extend a line at each vertex So. Interior + Exterior =

8 Geometry: Polygons (cont.) III. Similar Polygons When: Corresponding angles Ratios of the lengths of the sides are EQUAL! ie., A = J, B = K, C = L, and D = M 8 = 4 = 2 = IV. Similar Triangles When: Two angles (ie., B = D and a = a ) Ratios of the lengths of the sides are EQUAL! Find x

9 Geometry: Parallelograms and Trapezoids I. A parallelogram is a special quadrilateral where: 1. Opposite sides are parallel (AB DC, BC AD) 2. Opposite sides are equal length (AB = DC, BC = AD) 3. Opposite angles are equal (a = c, b = d ) 4. Consecutive angles are supplementary (a + b = b + c = c + d = 180, etc.) II. A trapezoid is a special quadrilateral where: 1. Only one pair of sides is parallel (b 1 b 2 ) = BASES 2. Area = ½ (b 1 + b 2 ) h Find the Area

10 Geometry: Circles I. Area II. Circumference A= π r 2 C = 2πr = Revolution **See a curve find the radius!!** vs. r = Easy! 1. Draw a diameter : Look up ratio 3. So...r =

11 Geometry: Circles (cont.) I. A = πr 2 II. C = 2πr **See a curve find the radius!!** III. Angles, Arcs, and Sectors Arc (think Length) Sector (think Area) 1. Find the proportion = n Solve for A (sector) or C (arc length) You try! 1. Find the area of the sector. 2. Find the length of minor arc AB. 3. Find the distance an ant travels along the circle Through how many degrees does the minute hand of a clock move from 1:25 P.M. to 1:37 P.M. of the same day? 11

12 Geometry: Equation of a Line The first thing to notice when you look at a line is the! This tells you about the steepness of the line. Slope = Rise over run, Change in y over change in x Example: Find the slope of line p, which contains the points (6, 4) and (0, -4). 1. The slope of line is + 2. The slope of line is - If the slope of a line is 3, find the following: Parallel m = NOT b Slope y-intercept (b, the coordinate value when x = 0) Perpendicular m = anything Reflection m = - b (if about x-axis) b (if about y-axis) If point B has the coordinates (1, p), then a perpendicular line has what slope? 12

13 Geometry: Equation of a Line cont. Now let s use the slope and y-intercept to write the equation of a line. There are three ways to do this, but two forms will be most useful to you on the SAT: Slope-intercept & Point-slope. Use reasoning to decide which form is best for the problem and when in doubt, use what you are most comfortable with! I. Slope-intercept: y = mx + b m = slope b = y-intercept Example: Write the equation of the line shown on the right. 1. Write the formula: y = mx + b 2. Pick two points and find the slope. 3. Find the y-intercept. 4. Plug-in! II. Point-slope: (y-y 1 ) = m(x-x 1 ) m = slope (x, y) and (x 1, y 1 ) = the coordinates of two points on the line Helpful when you cannot see the y-intercept or are given two points and a slope. Example: Wait! What if you are given the equation of a line in a different form??? Example: What is the slope of the line that is perpendicular to the line whose equation is 2y + 3x = 6? Solve for y! This makes it easy to identify m. 13

14 Equation of a Line Practice Problems 1. Write the formula! 2. Identify m and b, if necessary. How? 1. Use the graph. 2. Plug-in x and y

15 Geometry on the coordinate plane Don t be intimidated when you see a shape in the x/y coordinate plane. Draw a picture. Write the formula. Now use the points to help you identify information about the problem!

16 Geometry on the coordinate plane (cont.) Don t be intimidated when you see shapes in the x/y coordinate plane. Draw a picture. Write the formula. Now the points will help you identify information about the problem!

17 Geometry: Common Functions and Shifts Quadratic y = ax 2 + bx + c if a is +, parabola opens up vertex is the lowest point if a is -, parabola opens down vertex is the highest point Absolute Value y = x Exponential y = b x if b > 1, graph increases to the right if 1 > b > 0, graph decreases to the right *if b is a positive number other than 1, the graphs of y = b x and (1/b) x will be mirror images of each other about the y-axis. 17

18 Geometry: Transformations and Shifts I. Transformations Start with y = f(x) Replace f(x) with f(x) Replace x with - x Switch x and y Effect Reflect over the x-axis Reflect over the y-axis Reflect over the line y = x II. Shifts Start with y = f(x) y = f (x-h) y = f ( x + h) y = f(x) + k y = f(x) - k Effect shift h units RIGHT shift h units LEFT shift k units UP shift k units DOWN III. Stretching/Shrinking Start with the function y = c * f(x) c > 1 stretches the graph c < 1 shrinks the graph 18

19 Geometry: Common Functions and Shifts Practice