GEOMETRY: CHAPTER 2

1. GEOMETRY: CHAPTER 2 Ch. 2.1 Conditional Statements

2. A conditional statement is a logical statement that has two parts, a hypothesis and a conclusion. When a conditional statement is written in if-then form , the “if” part contains the hypothesis and the “then” part contains the conclusion. Ex.1 If it is raining, then there are clouds in the sky. Hypothesis Conclusion

3. Ex. 2 Rewrite the statements in if-then form. • All birds have feathers. • Two angles are supplementary if they are a linear pair. • The measure of two angles added together is 90 degrees, and the angles are complementary. a. If an animal is a bird, then it has feathers. b. If two angles are a linear pair, then they are supplementary.

4. Ex. 2. (cont.) Rewrite the statements in if-then form. c. The measure of two complementary angles added together is 90 degrees. If two angles are complementary, then their measures add up to 90 degrees.

5. Ex. 3. Rewrite the conditional statement in if-then form. 3x + 2 = 8, because x= 2

6. Ex. 3. (cont.)Rewrite the conditional statement in if-then form. 3x + 2 = 8, because x= 2 If x= 2, then 3x + 2 = 8

7. The negation of a statement is the opposite of the original statement. Notice that Statement 2 is already negative, so its negation is positive. Statement 1 —The sky is overcast. Negation — The sky is not overcast. Statement 2 —The ball is not Abby’s. Negation — The ball is Abby’s.

8. Conditional Statements can be true or false. To show that a conditional statement is true, you must prove that the conclusion is true every time the hypothesis is true. To show that a conditional statement is false, you need to give only one counterexample.

9. To write the converse of a conditional statement, exchange the hypothesis and conclusion. To write the inverse of a conditional statement, negate both the hypothesis and the conclusion. To write the contrapositive, first write the converse and then negate both the hypothesis and conclusion.

10. Ex. 4 Conditional, Converse, Inverse, Contrapositive

11. Ex. 5. Write the if-then form, the converse, the inverse, and the contrapositive of the conditional statement “Mission students are female.” Decide whether each statement is true or false.

12. Ex. 5. (cont.) If-then form: If a student attends Mission, then she is female. TRUE Converse: If a student is female, then she attends Mission. FALSE. Inverse: If a student does not attend Mission, then the student is not female. FALSE Contrapositive: If a student is not female, the student does not attend Mission. TRUE.

13. A conditional statement and its contrapositive are either both true or both false. Similarly, a converse inverse of a conditional statement are either both true or both false. When two statements are both true or both false, they are called equivalent statements.

14. Taken from: http://z.about.com/d/math/1/0/M/2/perpendicular.jpg KEY CONCEPT—PERPENDICULAR LINES Definition: If two lines intersect to form a right angle, then they are perpendicular lines. The definition can also be written using the converse: If two lines are perpendicular lines, then they intersect to form a right angle. You can write “line l is perpendicular to line m”as

15. Taken from: http://education.yahoo.com/homework_help/math_help/solutionimages/minigeogt/2/1/1/minigeogt_2_1_1_25_10/f-10-24-1.gif Ex. 6. Decide whether each statement about the diagram is true.

16. Point, Line, and Plane Postulates: Postulate 5—Through any two points there exists exactly one line. Postulate 6—A line contains at least two points. Postulate 7—If two lines intersect, then their intersection is exactly one point. Postulate 8—Through any three noncollinear points there exists exactly one plane.

17. Postulate 9—A plane contains at least three noncollinear points. Postulate 10—If two points lie in a plane, then the line containing them lies in the plane. Postulate 11—If two planes intersect, then their intersection is a line.

18. Ex.7. Solution: • Postulate 7—If two lines intersect, then their intersection is exactly one point. • Postulate 11—If two planes intersect, then their intersection is a line. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.

19. Ex. 8: Use the diagram to write examples of Postulates 9 and 10. Postulate 9—Plane P contains at least three noncollinear points A, B, and C. Postulate 10—Point A and point B lie in the same plane P, so line n containing A and B also lies in plane P. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.

20. Extra Ex. 9: Use the diagram to write examples of postulates 6 and 8. Postulate 6: Line l contains at least two points R and S. Postulate 8: Through noncollinear points R, S, and W, there exists exactly one plane M. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.

21. Concept Summary Interpreting a Diagram When you interpret a diagram, you can assume information about size or measure only if it is marked. You can assume All points shown are coplanar That angle AHB and angle BHD are a linear pair. That angle AHF and BHD are vertical angles. A, H, J and D are collinear. Line AD and line BF intersect at H. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.

22. Concept Summary (cont.) You cannot assume: Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 97.

23. Perpendicular Figures—A line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. In a diagram, a line perpendicular to a plane must be marked with a right angle symbol. Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 98.

24. Ex. 10 Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 98.

25. Images used for this presentation came from the following websites: http://z.about.com/d/math/1/0/M/2/perpendicular.jpg http://education.yahoo.com/homework_help/math_help/solutionimages/minigeogt/2/1/1/minigeogt_2_1_1_25_10/f-10-24-1.gif