Chapter 1 Basics of Geometry

1. Chapter 1 Basics of Geometry

2. 1.1 Notes Patterns & Inductive Reasoning Example 1: Vocabulary A conjecture is an unproven statement that is based on observations. Inductive reasoning is a process that includes looking for patterns and making conjectures. A counterexample is an example that shows a conjecture is false.

3. Example 2: Describing a Visual Pattern Directions: Sketch the next figure in the pattern.

4. Example 3: Describing a Number Pattern Directions: Describe a pattern in the sequence of numbers. Predict the next number. Subtract 1 to get the second number, then subtract 2 to get the third number, then subtract 3 to get the forth number. To find the fifth number, subtract 4 from the fourth number. So the next number is -5. Each number is one half the previous number. The next number is 8.

5. Example 4: Completing a Conjecture Directions: Complete the following conjecture by experimenting with some different math problems. A. Conjecture: The sum of any two odd numbers is ______________. Show work:

6. Example 5: Finding a Counterexample Directions: Show the conjecture is false by find a counterexample. A. Conjecture: If the difference of two numbers is odd, then the greater of the two numbers must also be odd. Show work by using a counterexample:

7. 1.2 Notes Points, Lines, & Planes Example 1: Vocabulary

10. Example 2: Naming Collinear and Coplanar Points

11. Example 3: Drawing Lines, Segments, & Rays B C A D Directions: Follow the directions below using the noncollinear points A, B, C, & D. Use a ruler.

12. Example 4: Identifying Opposite Rays

13. 1.3 Notes Segments & Their Measures Example 1: Vocabulary Rules that are accepted without proof. A real number that corresponds to a point on a line. A statement that can be demonstrated to be true by accepted mathematical operations and arguments. The process of showing a theorem to be correct is called a proof.

14. Example 1: Vocabulary Continued… The distance between 2 points on a line is the absolute value of the difference between the coordinates of the points. Segments that have the same length. AB = AD AB ≌AD

15. Example 2: Ruler Postulate

16. Example 3: Finding the Distance Between Two Points

17. Example 4: Segment Addition Postulate AB + BC = AC

18. Example 5: Applying Segment Addition Postulate

19. Example 6: The Distance Formula

20. Example 7: Applying the Distance Formula Directions: Use the points given to find the distance between the points by using the distance formula.

21. Example 8: Graphing Coordinate Points & Using the Distance Formula Directions: Graph A(-1,1), B(-4, 3), C(3,2), & D(2,-1). Connect B to D & A to C. Find the distance of AB, AC, AD, & BD by using the Distance Formula.

22. 1.4 Notes Angles & Their Measures Example 1: Vocabulary Two rays that have the same initial point The initial point of the angle A point is in between points that lie on each side of the angle A point not on the angle or in its interior.

23. Angles that have the same measure. Measures less than 90º Measures exactly 90º Measures more than 90º

24. Measures exactly 180º 2 angles are adjacent angles if they share a common vertex and side, but have no common interior points.

25. Example 2: Naming Angles P Q S R A B S T C U R

26. Example 3: Protractor Postulate

27. Example 4: Using the Protractor Postulate Directions: Use the diagram to find the measure of the angle.

28. Example 5: Angle Addition Postulate

29. Example 6: Applying the Angle Addition Postulate

30. Example 7: Classifying & Measuring Angles Directions: State whether the angle appears to be acute, right, obtuse, or straight. Then use a protractor to find its measure.

31. 1.5 Notes Segment & Angle Bisectors Example 1: Vocabulary The point that divides, or bisects, a segment into two congruent segments. To bisect a segment or an angle means to divide it into two congruent parts. A segment, ray, line, or plane that intersects a segment at its midpoint. A ray that divides an angle into two adjacent angles that are congruent.

32. Example 2: The Midpoint Formula

33. Example 3: Using the Midpoint Formula

34. Example 4: Finding an Endpoint

35. Example 5: Dividing an Angle Measure in Half

36. Example 6: Finding the Measure of an Angle Directions: Find the value of x. C.is continued on next slide…

37. Directions: Find the value of x.

38. 1.6 Notes Angle Pair Relationships Example 1: Vocabulary Consist of two angles whose sides form two pairs of opposite rays. Consists of two adjacent angles whose noncommon sides are opposite rays. The sum of the measures of an angle and its complement is 90º. Supplementary angles are two angles whose measures have the sum 180º.

39. Example 2: Identifying Vertical Angles & Linear Pairs 2 1 4 3 Are ∠2 and ∠3 a linear pair? _____________ Are ∠3 and ∠4 a linear pair? _____________ Are ∠1 and ∠3 vertical angles? ___________

40. Example 3: Finding Angle Measures

41. Example 4: Finding the Complementary Angles Directions: The two angles are complementary. Find the measure of the missing angle.

42. Example 5: Finding the Supplementary Angles Directions: The two angles are supplementary. Find the measure of the missing angle.

43. 1.7 Notes Perimeter, Area, & Circumference Example 1: Perimeter, Area, & Circumference Formulas

44. Example 2: Perimeter & Area of a Square Directions: Find the perimeter and area of the squares. Identify your variables.

45. Example 3: Perimeter & Area of a Rectangle Directions: Find the perimeter and area of the rectangles. Identify your variables.

46. Example 4: Perimeter & Area of a Triangle Directions: Find the perimeter and area of the triangles. Identify your variables.

47. Example 5: Perimeter & Area of a Circle Directions: Find the perimeter and area of the circles. Identify your variables.