Y. Davis Geometry Notes Chapter 5

Perpendicular Bisectors A segment, ray, line or plane that is perpendicular to a segment at its midpoint.

Theorem 5.1 Perpendicular Bisector Theorem If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Theorem 5.2 Converse of Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.

Concurrent Lines 3 or more lines that intersect at the same point.

Point of Concurrency The point where 3 or more lines intersect.

Circumcenter The point of concurrency for perpendicular bisectors of a triangle.

Theorem 5.3 Circumcenter Theorem The perpendicular bisectors intersect at a point called the circumcenter, that is equidistant from the vertices of the triangle.

Theorem 5.4 Angle Bisector Theorem If a point lies on the angle bisector, then it is equidistant from the sides of the angle.

Theorem 5.5 Converse of the Angle Bisector Theorem If a point is equidistant from the sides of the angle, then it lies on the angles bisector.

Incenter The point of concurrency for the angle bisectors of a triangle.

Therorem 5.6 Incenter Theorem The angle bisector of a triangle intersect at a point called the incenter, that is equidistant from the sides of the triangle.

Medians Is a segment with endpoints on the vertex of a triangle and the midpoint of the opposite side.

Centroid The point of concurrency for the medians of a triangle.

Theorem 5.7 Centroid Theorem The medians of a triangle intersect at a point called the centroid that is two-thirds of the distance from each vertex to the midpoint of the opposite side.

Altitude Is a perpendicular line from the vertex to the line containing the opposite side.

Orthocenter The point of concurrency for the altitudes of a triangle.

Equidistant from sides Concurrent Lines Point of Currency Facts Locations (Triangles) Perpendicular bisectors Circumcenter Equidistant from vertices Acute—Inside Right—Inside Obtuse--Inside Angle Bisectors Incenter Equidistant from sides Right—On(hypotenuse) Obtuse--Outside Medians Centroid 2/3 the length from vertex to midpoint Altitudes Orthocenter _____ Right—On (Right angle)

Inequality For all real numbers, a>b, if and only if there is a positive number c, so that a=b+c

Theorem 5.8 Exterior Angle inequality The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

Angle-Side Relationships in Triangles Theorem 5.9—In a triangle, the longest side is opposite the largest angle. The shortest side is opposite of the smallest angle. Theorem 5.10—In a triangle, the largest angle is opposite the longest side. The smallest angle is opposite the shortest side.

Indirect reasoning Assume that the conclusion is false and then show that the assumption led to a condradiction.

Indirect Proof Proof by contradiction. Assume the conclusion is false, by assuming the that the opposite is true. Use logical reasoning to show that this leads to a contradiction of the hypothesis, definition, postulate, theorem, or corollary. Since the assumption leads to a contradiction, then the original conclusion must be true.

Theorem 5.11 Triangle Inequality Theorem The sum of any two sides of a triangle must be greater than the length of the third side.

Theorem 5.13 Hinge Theorem If 2 sides of a triangle are congruent to 2 sides of another triangle, and the included angle of the 1st triangle is larger than the included angle of the 2nd triangle, then the 3rd side of the 1st triangle is longer than the 3rd side of the 2nd triangle.

Theorem 5.14 Converse of the Hinge Theorem If 2 sides of one triangle are congruent to 2 sides of another triangle, and the 3rd side of the 1st triangle is longer than the 3rd side of the 2nd triangle, then the included angle of the 1st triangle is larger than the include angle of the 2nd triangle.