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Geometry
Chapter 5 Notes
5.1
Perpendiculars and Bisectors
Theorem 5.1 – Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the
endpoints of the segment.
If CP is the perpendicular bisector of AB, then CA = CB
Theorem 5.2 – 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.
If DA = DB, then D lies on the perpendicular bisector
Theorem 5.3 – Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two sides of the
angle.
If m BAD = m CAD, then DB = DC
Theorem 5.5 – Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is equidistant from the sides of the angle,
then it lies on the bisector of the angle.
If DB = DC, then m BAD = m CAD
5.2
Bisectors of a Triangle
Definition – When two figures are congruent, there corresponding angles are
congruent and their corresponding sides are congruent
Theorem 5.5 – Concurrency of Perpendicular Bisectors of a Triangle
The perpendicular bisectors of a triangle intersect at a point that is equidistant from
the vertices of the triangle.
PA = PB = PC
Theorem 5.6 – Concurrency of Angle Bisectors of a Triangle
The angle bisectors of a triangle intersect at a point that is equidistant from the sides
of the triangle.
PD = PE = PF
5.3
Medians and Altitudes of a Triangle
Theorem 5.7 – Concurrency of Medians of a Triangle
The medians of a triangle intersect at a point that is two thirds of the distance from
each vertex to the midpoint of the opposite side
If P is the centroid of ABC , then
AP = ⅔AD, BP = ⅔BF, CP = ⅔CE
Theorem 5.8 – Concurrency of Altitudes of a Triangle
The lines containing the altitudes of a Triangle are concurrent
If AE, BR, and CD are the altitudes of
ABC, then the lines AE, BF, and CD
Intersect at some point H.
5.4
Midsegment Theorem
Theorem 5.9 – Midsegment Theorem
The segment connecting the midpoints of two sides of a triangle is parallel to the
third side and is half as long
DE || AB and DE = ½ AB
5.5
Inequalities in One Triangle
Theorem 5.10 –
If one side of a triangle is longer than another side, then the angle opposite the longer
side is larger than the angle opposite the shorter side.
Theorem 5.11 –
If one angle of a triangle is larger than another angle, then the side opposite the
larger angle is longer than the side opposite the smaller angle.
Theorem 5.12 – Exterior angle Inequality
The measure of an exterior angle of a triangle is greater than the measure of either
of the two nonadjacent interior angles.
m  1 > m A and m  1 > m B
Theorem 5.13 – Triangle Inequality
The sum of the lengths of any two sides of a triangle is greater than the length of the
third side.
AB + BC > AC
AC + BC > AB
AB + AC > BC
5.6
Indirect Proof and Inequalities in Two Triangles
Theorem 5.14 – Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the
included angle of the first is larger than the included angle of the second then the
third side of the first is longer than the third side of the second.
RT > VX
Theorem 5.15 – Converse of the Hinge Theorem
If two sides of one triangle are congruent to two sides of another triangle, and the
third side of the first is longer than the third side of the second, then the included
angle of the first is larger than the included angle of the second.
m  A > m D