1. 4-1 Using Logical Reasoning

2. The Conditional The conditional, also know as the “if-then statement” consists of two parts. (The hypothesis and the conclusion.) In basic terms, the hypothesis comes after “if” and the conclusion comes after “then.” Ex: If you want to hit a homerun, then buy the Easton Stealth.

3. Truth Value When you determine a conditional true or false, you determine its truth value. To prove a conditional false, one must find a counterexample. (This is just something that proves a conditional false.) Ex: Find a counterexample for the following: Every student has a B average in biology.

4. Truth Value (cont.) Counterexample: Jamie Lee has an A average.

5. The Converse The Converse statement interchanges the order of the hypothesis and conclusion with the exception of the words “if” and “then.” Ex: Conditional: If a polygon has six sides, then it is a hexagon. Converse: If a polygon is a hexagon, then it has six sides.

6. Biconditional When a conditional and a converse are true, they can be combined together as a biconditional by using the words “if and only if between the conditional and converse. Conditional: If a polygon has six sides, then it is a hexagon. Converse: If a polygon is a hexagon, then it has six sides. Biconditional: A polygon is a hexagon if and only if it has six sides.

7. Negation A negation of a statement has the opposite meaning. Ex: Pookie has black fur. Negation: Pookie does not have black fur.

8. The Inverse The inverse of a conditional negates both the hypothesis and conclusion. Ex: Conditional: If Miss Oliver does not take her creatine, then she is not moody. Inverse: If Miss Oliver takes her creatine, then she is moody.

9. The Contrapositive The contrapositive of a conditional interchanges and negates the hypothesis and the conclusion. Ex: Conditional: If a figure is a square, then it is a rectangle. Contrapositive: If a figure is not a rectangle, then it is not a square.,

10. 4-2 Isosceles Triangles

11. The Structure of an Isosceles Triangle An isosceles triangle always has two congruent sides, or legs which form a vertex angle. (angle A) The two bottom angles are called base angles, which are congruent angles located on the base.( segment BC)

12. (cont.) If segments CA and CB are congruent and segment MC bisects angle BCA, then segment MC is the perpendicular bisector of the base, thus, forming two supplementary right angles.

13. 4-3 Preparing For Proof A proof is a statement or diagram that proves something to be true. Ex: A drivers license proves that a person is allowed to drive. A two-column proof proves statements true with two columns: a statements section and a section for the reasons that prove the statements true.

14. Two Column Proof A two column proof always has given information. In this case, (geometry) a side or angle of a figure can be given.

15. Example of a Two Column Proof

16. 4-4 Midsegments of Triangles A midsegment of a triangle is a segment connecting the midpoints of two of its sides. If any segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and half its length.

17. Examples of Midsegments on triangles If segment BC is 5 meters long, segment DE is 2.5 meters long because of the Triangle Midsegment Theorem.

18. 4-5 Using Indirect Reasoning Indirect Reasoning- all possibilities are considered and then all but one are proved false. Step 1: Assume the opposite of what you want to prove true. Step 2: Reach a contradiction of an earlier statement. State that the assumption made was false. Step 3: State that what you wanted to prove must be true.

19. Example: Use indirect reasoning for the following statement: The lake is starting to freeze over. Show that the temp. outside must be 32 degrees F or less. 1. 2. 3.

20. 4-6 Triangle Inequalities The sum of the lengths of 2 sides of a triangle is greater than the length of the third. A B C 57 10 Ex: AB + AC > BC The longer side lies opposite the larger angle and vice versa. A B C 20 120 40 14cm 9cm 12cm Ex: Listed largest to smallest. <B across from AC <C across from AB <A across from BC

21. 4-7 Bisectors and Locus A point on the perpendicular bisector of a segment is equidistant from the ends of the segment and vice versa. A B C D

22. Locus- set of points that meet a stated condition. Ex: Sketch the locus of all points in a plane 3cm from segment MN. MN

23. 4-8 Concurrent Lines Concurrent- when 3 or more lines intersect in one point. Point of Concurrency- the point at which concurrent lines meet. Point of concurrency

24. The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. (same with angle bisectors) A B CD E F

25. Median- segment whose endpoints are a vertex and a midpoint of the side opposite the vertex. Altitude- perpendicular segment from a vertex to the line opposite the vertex. Angle Bisector- segment that cuts and angle in half and extends down to the segment opposite the angle. Perpendicular Bisector- segment that starts at a vertex and extends down to the side opposite the vertex, forming a 90 degree angle. *Note: Medians of triangles are concurrent.

26. Perpendicular BisectorAngle Bisector MedianAltitude