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# Chapter Five

### Relationships in Triangles

TermDefinition
perpendicular bisector In a triangle, a line segment, or ray that passes through the midpoint of a side and is perpendicular to that side.
concurrent lines three or more lines intersecting at a common point.
point of concurrency the point where concurrent lines intersect
circumcenter the point of concurrency of the perpendicular bisectors
incenter the point of concurrency for the angle bisectors
Perpendicular Bisector 5.1 Theorem If a point is on a perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
Circumcenter Theorem The perpendicular bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle.
Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.
Incenter Theorem The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from each side of the triangle.
median a triangle of a segment with endpoints being a vertex of a triangle and the midpoint of the opposite side.
altitude a segment from a vertex to the line containing the opposite side and perpendicular to the line containing that side.
centroid the point of concurrency of the medians of a triangle
orthocenter the lines containing the altitudes of a triangle are concurrent and intersect at the orthocenter.
Angle Side Relationships in Triangle If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.
Angle Side Relationship 5.10 If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Created by: amgeometry