1. MAT 105 SPRING 2009
Chapter 2
Geometry

2. Section 2.1 Lines and Angles
MAT 105 SPRING 2009
Section 2.1 Lines and Angles

3. The Foundation Point: Usually named using a capital letter
Line: (1-dimensional) Usually named using two points included on the line
Plane: (2-dimensional)
Ray: (half-line) Has one endpoint and extends infinitely in one direction A B A B

4. Angles
An angle is formed by two ___________ with a common _____________, called the ____________ of the angle.
The amount of ________________ of the terminal ray is called the angle. A C B

5. Types of Angles Right Angle ________________
Straight Angle ________________
Acute Angle ________________
Obtuse Angle _________________

6. Types of Lines
Lines in the same plane that do not intersect are called ____________________ lines.
Lines that intersect at right angles are called _______________________ lines. B A
AB CD D C

7. More on Angles
________________________ angles are two angles whose sum is 90°.
________________________ angles are two angles whose sum is 180°.
________________________ angles share a common vertex and a common side. E F G H

8. _______________ angles are equal in measure (congruent).
When two lines intersect, the angles that are formed on opposite sides of the point of intersection are called ___________________ angles.
_______________ angles are equal in measure (congruent). 2 1 3 4

9. When two parallel lines are cut by a transversal, the
If a line intersects two or more lines in a plane, it is called a _____________________________. M N P Q
When two parallel lines are cut by a transversal, the
corresponding angles are equal (congruent)
alternate interior angles are equal (congruent)
alternate exterior angles are equal (congruent)

10. Pairs of alternate interior angles: M N P Q 1 3 2 4 5 6 7 8
Pairs of corresponding angles: ________&_________; ________&_________; ________&_________; ________&_________
Pairs of alternate interior angles:
________&_________; ________&_________
Pairs of alternate exterior angles:

11. If 4 = 41°, find the measures of the remaining angles.P Q 1 3 2 4 5 6 7 8
Example
If 4 = 41°, find the measures of the remaining angles.
1 =______ 2 = ______ 3 = ______
5 =______ 6 = ______ 7 = ______
8 =______

12. Ratios of corresponding segments of the transversal are equal.
When more than two parallel lines are cut by two transversals, the segments of the transversals between the same two parallel lines are called corresponding segments. A B C D E F W X Y Z
Ratios of corresponding segments of the transversal are equal.

13. Example p 52 # 23-28 Find the measures of the angles. 1) BDF =
2)  ABE =
3)  DEB =
4)  DBE =
5) DFE =
6) ADE =
44° A B C D E F

14. Section 2.2 Triangles

15. Polygons A polygon is a closed figure with straight sides.
Some common polygons:
3 sides: __________________
4 sides: __________________
5 sides: __________________
6 sides: __________________
7 sides: __________________
8 sides: __________________
9 sides: __________________
10 sides: __________________
12 sides: __________________

16. Polygons
Perimeter: _______________________________________________________________________________________________________________________________________
Units used for perimeter:
Area: __________________________________________________________________________________________
Units used for area:

17. Triangles
Triangles are polygons with three sides and three interior angles.
The sum of the measures of the three angles of a triangle is __________.
Triangles are often classified according to the types of sides or the types of angles they contain.

18. Types of Triangles (classified by sides)
Scalene:
Isosceles:
Equilateral:

19. Right Triangles Pythagorean Theorem:
leg
hypotenuse
Pythagorean Theorem:
In a right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides (legs).
Example: If a = 5 and c = 11, find the exact value of b. c a b

20. Area of a Triangle
The area of a triangle is one-half the product of a base and its height.
The height (or altitude) of a triangle is the line segment drawn from a vertex perpendicular to the opposite side (base). b h
Area of a triangle:

21. Find the perimeter and area of the triangle shown below (all numbers are approximate).
6.5 cm
1.3 cm
2.2 cm
4.8 cm

22. Find the perimeter and area (to 2 s. d
Find the perimeter and area (to 2 s.d.) of an isosceles triangle whose two equal sides measure 35 mm and height from the vertex angle to the base measures 28 mm.

23. A Super Hero!
If we know the three lengths of the sides of a triangle, but not the height, we can use Hero’s Formula (Alternate: Heron’s Formula) to calculate the area.
Hero’s Formula

24. Example
A triangular-shaped park is bounded by three streets. The lengths of the three sides of the park are found to be 358 ft, 437 ft, and 509 ft. What is the area of the park (to 3 sig dig)?

25. Congruent Triangles
Two or more triangles are congruent if the measures of each of the ___________ and the measures of each of the ____________ are the same. A B C
84
55
41
5.8 cm
7.0 cm
4.6 cm P Q R
84
55
41
5.8 cm
7.0 cm
4.6 cm

26. Similar Triangles
Similar triangles have congruent ________________ but do not necessarily have congruent _______________.
Properties of Similar Triangles
Corresponding angles of similar triangles are equal.
Corresponding sides of similar triangles are proportional. A B C D E
ABC  ADE
Corresponding angles:
Corresponding sides:

27. Example
A woman is standing next to a building on level ground. The woman, who is 5’6” tall, casts a shadow that is 3.0 feet long. At the same time, a building casts a shadow that is 92 feet long. How tall is the building? Round ans to 2 sig digits.

28. Section 2.3 Quadrilaterals

29. A quadrilateral is a polygon with ______ sides and _____ interior angles.
The sum of the measures of the interior angles of a quadrilateral is ___________.
A ________________ of a polygon is a line segment joining any two non-adjacent vertices.

30. Special Types of Quadrilaterals
Parallelogram
Rhombus
Rectangle
Square
Trapezoid

31. Determine if each statement is true sometimes, always, or never.
A rectangle is a parallelogram. _______________
A rhombus is a square. _______________
A square is a rhombus. _______________
A trapezoid is a parallelogram. _______________
A square is a rectangle. _______________
A parallelogram is a rectangle. _______________
A square is a trapezoid. _______________

32. Perimeter
The perimeter of a quadrilateral is the sum of the lengths of the four sides. (The distance around the figure.)
Some Special Formulas:
Perimeter of a Rectangle: P = ________________
Perimeter of a Square: P = ____________

33. Area Formulas Area of a Parallelogram: A = ________________
Area of a Rectangle: A = ________________
Area of a Square: A = ________________
Area of a Trapezoid: A = ________________

34. Examples
1) Text p. 62 # 30

35. Examples
2) Text p. 62 # 32

36. Examples
3) Text p. 63 # 38

37. Section 2.4 Circles

38. Some Circle Vocab
________________: The set of all points equidistant from a fixed point, called the __________.
________________: A line segment with endpoints on the circle that passes through its center.
________________: A line segment from the center to a point on the circle.
________________: A line segment with endpoints on the circle.

39. More Circle Vocab
________________: A line that intersects (touches) the circle at EXACTLY ONE POINT.
________________: A line that passes through two points of the circle. o
The radius is perpendicular to the tangent at the point of tangency.

40. An example If MNO = 14°, find MOP.
Let O be the center of the circle.
MN is tangent to the circle. N M o P
If MNO = 14°, find MOP.

41. Circumference Circumference of a Circle:
 (pi) is an irrational number that is approximately equal to ______
 is the ratio of the circumference (perimeter) of a circle to its diameter.
Circumference of a Circle:

42. Area Area of a Circle: Example
Find the area of a circle with diameter 18.5 cm. Give answer to three significant digits.

43. An example
A circular walkway, with a uniform width of 3.0 ft, is to be installed around a traffic circle that has a diameter of 18 ft. Find the area of the walkway to two significant digits.

44. Another example
What is the area of the largest circle that can be cut from a rectangular plate 21.2 cm by 15.8 cm? How much waste is there? Give answers to 3 sig digits.
21.2 cm
15.8 cm

45. One More Example
Neglecting waste, how much would it cost to lay down a hardwood floor on the indoor rink if flooring costs $22.50 per square meter? Round to nearest dollar.
Note: Ends are semicircular.
14 m
20 m

46. Arcs & Angles
A _________ __________ is an angle formed by two radii (its vertex is the center of the circle.)
An _________ is a part of the outside of the circle (a curved segment) A B o
The measure of an arc is the same as the measure of the central angle that forms it. (Measured in degrees.)

47. Regions of a Circle
A ___________________ is a region bounded by two radii and the arc they intercept.
A ____________ is a region bounded by a chord and its arc. A B o A B

48. More Angles An ____________ ___________ has its vertex on the circle.B
Inscribed angle
Intercepted arc
The measure of an inscribed angle is half of its intercepted arc.

49. An Example
If the measure of an inscribed angle is 45°, find the length (to 2 sig dig) of its intercepted arc, given that the radius of the circle is 8.0 cm.

50. Section 2.6 Solid Geometric Figures
MAT 105 SPRING 2009
Section 2.6 Solid Geometric Figures

51. Volume & Surface Area Units used for surface area:
Volume is the measure of space occupied by a solid geometric figure.
Units used for volume:
Surface Area is the total area of all of the faces of a solid geometric figure.
Units used for surface area:

52. Formulas for Volume & Surface Area
V = Volume
A = Total Surface Area
S = Lateral Surface Area (does not include area of the bases)

53. Solid Geometric Figures

54. Formulas for Volume & Surface Area
Solid Geometric Figure
Volume Formula
Surface Area Formula
Rectangular Solid
Cube
Right Circular Cylinder
Right Circular Cone
Regular Pyramid
Sphere
e is length of edge of cube s is the lateral height
B is area of the base p is the perimeter of the base

55. Example 1
P 74 # 26

56. Example 2
P 74 # 28

57. Example 3
P 74 # 34