1. Welcome to our sixth seminar! We’ll begin shortly

2. Line segments A point is a spot in space that has no length, width or height. Think of it as a place holder. A line segment is a line that ends at two points. For example: A B This line segment is named AB or AB

3. Rays Rays are lines with only one end point. You can think of them as starting at a point and then extending into infinity in one direction. For example: C D This ray is called AB or AB

4. Angles An angle is made up of two rays, lines, or line segments which start at the same endpoint (called the vertex). For example: A B CR ST J K L M This angle is named ے ABC, ے CBA, ے B or ے k km a This angle is named ے RST, ے TSR, ے S or ے m The highlighted angle is named ے JNL, ے LNJ, or ے a N

5. Types of angles 180 0 90 0 360 0 < 90 0 > 90 0 Line segment One full rotation= 360 0 Straight angle = 180 0 Right angle = 90 0 An acute angle is one that measures less than 90 0 An obtuse angle is one that Measures more than 90 0

6. More types of angles Angles whose sum is 90 0 are called complementary angles. Examples: Angles whose sum is 180 0 are called supplementary angles 60 0 30 0 58 0 32 0 134 0 46 0 60 0 +30 0 =90 0 32 0 +58 0 =90 0 46 0 +134 0 =180 0 Note that if you know one angle you can calculate the other: 38 0 The sum of a and 38 is 180 a + 38 = 180 Subtract 38 from both sides: a = 180 – 38 a = 142 0 a

7. Planes (a 2 dimensional surface)

8. Pairs of lines Parallel lines have the same distance between them at each point. They never intersect and the angle between them is 0 0 Perpendicular lines intersect at 90 0. Intersecting lines meet at a point at an angle not 90. The vertical angles (opposite) are equal to each other. The sum of the adjacent angles (next to each other) is 180. Here Vertical angles are a,c, and b,d. Adjacent angles are a,b; b,c; c,d; and a,d. a b c d ab cd ef g h A transversal line is one that intersection two other lines. Corresponding angles are those on the same side of the Two lines. Here they are a,e; c,g; b,f; and d,h Alternate interior angles are those that are opposite Interior. Here they are c,f and d,e. If the two lines are parallel then Corresponding lines are equal and alternate interior angles Are also equal.

9. A few examples; solve for all angles 35 0 122 0 ab c d ef hi c and 35 are vertical angles and equal so c = 35 0 a and c are adjacent angles so a + c= 180 0 a +35 = 180 0 a = 180 0 -35 0 a = 145 0 and b = 145 0 (opposite) Summary: a = 145 0, b = 145 0, c =35 0 122 0 and i are corresponding angles; i = 122 0 122 0 and e; I and j are vertical angles; e=122 0, j=122 0 d and 122 0 are adjacent angles so d + 122 0 = 180 0 d = 180 0 – 122 0 d = 58 0 d and h are vertical angles so h = 58 0 d, f and h,k are vertical angles so f=58 0 and k = 58 0 Summary: d=58 0, e=122 0, f=58 0, h=58 0, i=122 0, j=122 0,k = 58 0 j k

10. Polygons (closed 2-D figures)

11. Types of triangles

12. Similar figures (same shape, different sizes)

13. Example

14. Area and perimeter formulas

15. Perimeter The perimeter (P) of a polygon (a two-dimensional shape with at least three sides) is the sum of all of the sides. In other words it is the distance around the shape. Don’t forget to include units; they will be in length (like m, ft, in, etc). What is the perimeter of this trapezoid (a four sided shape with unequal sides)? P = the sum of the sides P = 28 + 15 + 8 + 12 P = 43 + 20 P = 63 units NOTE: we ignored the extraneous information (the height). If there are no given units write “units”

16. A dollar bill has a width of 2.56in and a length of 6.14 inches. Find the perimeter.

17. Example

18. This figure contains a rectangle which is 8 by 3 ft and a triangle with a height of 8 ft and a base of 6 – 3 or 3ft. Add these two areas to get the total

19. Volume and formula problems V=πr 2 h

20. Surface area: the total amount of area on the surface of a three dimensional figure. The units are the same as area: length squared. Here are some of the common formulas for finding surface area:

21. Explaining the cylinder Here is a picture of the surfaces of a cylinder that is enclosed. I has the area of each end and the rectangle that surrounds it. If it was not enclosed or had only one end enclosed you remove those parts from the equation.

22. Examples Find the surface area of a closed cylinder with a height of 16 inches and a diameter of 12 inches. The radius is half of 12. r = 6 in. round to the nearest tenth.

24. Volume: the amount of ‘stuff’ enclosed in a three dimensional object. Here is an example of what it looks like. The units are length cubed.

26. Examples We found the surface area of this object, now let’s find the volume.

27. Pyramid This is a triangular pyramid that is 2000ft tall and each base length is 2500ft. What is the area?

28. Euler’s formula: # of vertices - # edges + #face = 2

29. Example: If #vertices = 11, # faces = 5, find the # edges # of vertices - # edges + #face = 2 11 - E + 2 = 2 13 - E = 2 13 - E – 13 = 2 – 13 -E = -11 E = 11 There are 11 edges

30. Converting square and cube units

31. Thank you for attending!