1. If-Then Statements; Converses
2-1
If-Then Statements; Converses

2. CONDITIONAL STATEMENTS are statements written in if-then form
CONDITIONAL STATEMENTS are statements written in if-then form. The clause following the “if” is called the hypothesis and the clause following “then” is called the conclusion.

3. Examples If it rains after school, then I will give you a ride home.
If you make an A on your test, then you will get an A on your report card.

4. CONVERSE
is formed by interchanging the hypothesis and the conclusion.

5. Examples
False Converses
If Bill lives in Texas, then he lives west of the Mississippi River.
If he lives west of the Mississippi River, then he lives in Texas

6. Counterexample An example that shows a statement to be false
It only takes one counterexample to disprove a statement

7. Biconditional
A statement that contains the words “if and only if”
Segments are congruent if and only if their lengths are equal.

8. Properties from Algebra
2-2
Properties from Algebra

9. Addition Property
If a = b, and c = d,
then a + c = b + d

10. Subtraction Property
If a = b, and c = d,
then a - c = b - d

11. Multiplication Property
If a = b,
then ca = bc

12. Division Property
If a = b, and c 0
then a/c = b/c

13. Substitution Property
If a = b, then either a or b may be substituted for the other in any equation (or inequality)

14. Reflexive Property
a = a

15. Symmetric Property
If a = b, then b = a

16. Transitive Property
If a = b, and b = c, then a = c

17. Distributive Property
a(b + c) = ab + ac

18. Properties of Congruence

19. Reflexive Property
DE  DE
 D   D

20. Symmetric Property If DE  FG, then FG  DE
If  D   E, then  E   D

21. Transitive Property If DE  FG, and FG  JK, then DE  JK
If D   E, and  E   F, then  D   F

22. 2-3
Proving Theorems

23. Midpoint of a Segment – is the point that divides the segment into two congruent segments

24. If a point M is the midpoint of AB, then
THEOREM 2-1
Midpoint Theorem
If a point M is the midpoint of AB, then
AM = ½AB and MB=½AB

25. BISECTOR of ANGLE– is the ray that divides the angle into two adjacent angles that have equal measure.

26. Angle Bisector Theorem
If BX is the bisector of ABC, then:
mABX = ½mABC and mXBC = ½ m ABC

27. A • X • C B •

28. Special Pairs of Angles
2-4
Special Pairs of Angles

29. COMPLEMENTARY
two angles whose measures have the sum 90º J
39º
51º K

30. two angles whose measures have the sum 180º
SUPPLEMENTARY
two angles whose measures have the sum 180º H
133º G
47º

31. VERTICAL ANGLES– two angles whose sides form two pairs of opposite rays.

33. Vertical angles are congruent
THEOREM 2-3
Vertical angles are congruent

34. 2-5
Perpendicular Lines

35. Perpendicular Lines– two lines that intersect to form right angles ( 90° angles)

36. 2-4 THEOREM
If two lines are perpendicular, then they form congruent adjacent angles.

37. 2-5 THEOREM
If two lines form congruent adjacent angles, then the lines are perpendicular.

38. 2-6 THEOREM
If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary

39. 2-6
Planning a Proof

40. Parts of a Proof A diagram that illustrates the given information
A list, in terms of the figure, of what is given
A list, in terms of the figure, of what you are to prove
A series of statements and reasons that lead from the given information to the statement that is to be proved

41. 2-7 THEOREM
If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.

42. 2-8 THEOREM
If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.

43. THE END