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Section 4-1
Using Logical Reasoning

Conditional- if then statement
if- hypothesis then- conclusion

Truth value- determines if entire statement is true or false
Converse- changes hypothesis and conclusion
Biconditional statement- both conditional and converse statements are true and can be combined to one staement using if and only if

Negation- opposite- not
Inverse- negates both the hypothesis and the conclusion
Contrapositive- conditional interchanges and negates both the hypothesis and the conclusion

Section 4-2
Isosceles Triagles

Isosceles Triangle- has two congruent legs, the third side is the base, two congruent angles, the third angle is the vertex Angle

Theorem 4-1 isosceles triangle: if two sides of a triangle are congruent, then the angle opposite those sides are congruent.
If line AC is congruent to line BC, then Theorem 4-2: the bisector of the vertex angle of an iscoceles triangle is the perpendicular bisector of the base
If line AC is congruent to line BC and line CD bisects
Theorem 4-3 Converse of Isosceles Triangle: If two angles of a triangle are congruent, then the sides opposite the angles are congruent.
If
Corollary to Isosceles Triangle Theorem: If a triangle is equilateral, then it is equiangular
Corollary to Converse isosceles Triangle theorem: If a triangle is equiangular, then it is equilateral

Section 4-3
Proofs

Theorem 4-4: if a triangle is a right triangle, then the acute angles are complementary

Theorem 4-5: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are Congruent
Theorem 4-6: all right angles are congruent. All equal to 90 degrees no more, no less
Theorem 4-7: if two angles are congruent and supplementary, then each is a right angle

Section 4-4
Midsegments of triangles

Midsegment of a triangle- a segment connecting midpoints of two of its sides

Theorem 4-8 triangle midsegment: Ff a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and half its length

Section 4-5
Using Indirect Reasoning

Indirect Reasoning-all possibilities are considered and then all but one are proved false (deduction)

Writing an Indirect Proof
1. Assume that the opposite of what you want to proof is true
2. use logical reasoning to reach a contradiction of an earlier statement, such as the given theorem
3. State that what you want to prove must be true

Section 4-6
Triangle inequalities

Theorem 4-9 triangle inequality: the sum of the lengths of any two sides of a triangle is greater than the length of the Third side

Theorem 4-10: if two sides of a triangle are not congruent, then the larger angle, lies opposite the longer sides.
Theorem 4-11: If two angles of a triangel are not congruent, then longer side lies opposite the larger sides.

Section 4-7 Bisectors & Locus
Theorem 4-12 perpendicular Bisector theorem

If a point is on a perpendicular bisector of a segment,then it is equidistant from the end poins of a segment, then it is on the 1 bisector of the segment.

Theorem 4-13 Converse of the bisect Theorem:IF a point is equidistant from the end points of the segment, then it is on the 1 bisector of the segment.

Locus - a set of points that meets a stated condition.

Section 4-7
Theorem 4-14 Angle bisector theorem

If a pont is on the bisector of an angle, then it equidistant from the sides of the angle.

Theorem 4-15 Converse of Angle Bisector Theorem

If a point in the interior of an angle is equidistant of an angle is equidistant from the sides of the angle, then it is on the bisector.


Section 4-8 Concurrent Lines
Conconcurrent when 3 or more lines intersect is one point.

Point of concurrency - location of intersection of concurrence

Theorem 4-16
Perpendicular bisectors of he sides of a triangle are concurrent @ a point equidistant from the vertices.

Theorem 4-17
The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

Median /Attidude of a triangle a median is the middle of a ^ A median of a triangle is a segment that pairs a side opposite that vertex.

Attitude
Determine the height an attitude of a triangle is a perpendicular segment from a vertex to the one containing the side

Opposite the vertex.

Theorem 4-18
The lines that containt the attitude is concurrent.

Theorem 4-19 The medians of a Triangle are concurrent.

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