1. Warm Up Complete each sentence. 1. ? points are points that lie on the same line. 2. ? points are points that lie in the same plane. 3. The sum of the measures of two ? angles is 90°. Collinear Coplanar complementary

2. Find the next item in the pattern. Example 1A: Identifying a Pattern January, March, May,... The next month is July. Alternating months of the year make up the pattern. Find the next item in the pattern. Example 1B: Identifying a Pattern 7, 14, 21, 28, … The next multiple is 35. Multiples of 7 make up the pattern. Find the next item in the pattern. Example 1C: Identifying a Pattern In this pattern, the figure rotates 90° counter-clockwise each time. The next figure is.

3. When several examples form a pattern and you assume the pattern will continue, you are applying inductive reasoning. Inductive reasoning is the process of reasoning that a rule or statement is true because specific cases are true. You may use inductive reasoning to draw a conclusion from a pattern. A statement you believe to be true based on inductive reasoning is called a conjecture. Complete the conjecture. Example 2A: Making a Conjecture The sum of two positive numbers is ?. The sum of two positive numbers is positive.

4. Complete the conjecture. Example 2B: Making a Conjecture The number of lines formed by 4 points, no three of which are collinear, is ?. Draw four points. Make sure no three points are collinear. Count the number of lines formed: The number of lines formed by four points, no three of which are collinear, is 6.

5. Example 3 Make a conjecture about the lengths of male and female whales based on the data. In 5 of the 6 pairs of numbers above the female is longer. Female whales are longer than male whales. Average Whale Lengths Length of Female (ft) 495150485147 Length of Male (ft) 4745444648

6. To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. To show that a conjecture is always true, you must prove it. A counterexample can be a drawing, a statement, or a number. Inductive Reasoning 1. Look for a pattern. 2. Make a conjecture. 3. Prove the conjecture or find a counterexample.

7. Example 4a For any real number x, x 2 ≥ x. Show that the conjecture is false by finding a counterexample. Let x =. 1212 The conjecture is false. Since =, ≥. 1212 2 1212 1414 1414 Example 4b Supplementary angles are adjacent. Show that the conjecture is false by finding a counterexample. The supplementary angles are not adjacent, so the conjecture is false. 23 ° 157 °

8. Example 4c The radius of every planet in the solar system is less than 50,000 km. Show that the conjecture is false by finding a counterexample. Planets’ Diameters (km) MercuryVenusEarthMarsJupiterSaturnUranusNeptunePluto 488012,10012,8006790143,000121,00051,10049,5002300 Since the radius is half the diameter, the radius of Jupiter is 71,500 km and the radius of Saturn is 60,500 km. The conjecture is false. Pg. 77 _____________________________________________________