Ch 5 Properties AND Attributes of Triangles – HOLT Geom
5-1 Perpendicular and Angle Bisectors
If a point is on the
If
perpendicular
bisector of a
segment, then it is
A
equidistant from the
endpoints of the
segment.
Then
l X
Converse of the
Perpendicular
Bisector
Theorem
If a point is
If
equidistant from the
endpoints of the
segment, then it is on
A
the perpendicular
bisector of the
Then
segment.
l X
Locus
The set of points that
satisfies a given
condition
Perpendicular
Bisector
Theorem
Y
Y
B
B
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 If a point in the
Angle Bisector interior of an angle is
Theorem
equidistant from the
sides of the angle,
then it is on the
bisector of an angle.
5-2 Bisectors of Triangles
Concurrent
3 or more lines
intersect at a point
Point of
The point where
Concurrency
these lines intersect
Circumcenter of The point of
the Triangle
concurrency of the 3
perpendicular
bisectors
Circumcenter
The circumcenter of
Theorem
a triangle is
equidistant from the
vertices of the
triangle.
Circumscribed A circle drawn
outside of a triangle
Incenter of the
Triangle
The point of
concurrency of the 3
triangle angle
bisectors
Incenter
Theorem
Inscribed
The incenter of a
triangle is equidistant
from the sides of the
triangle
The circle drawn
inside a triangle
5-3 Medians and Altitudes of Triangles
Median of A segment
a Triangle whose
endpoints are a
vertex of the
triangle and the
midpoint of the
opposite side
Centroid The points of
of a
concurrency of
Triangle
the medians of
AKA the
a triangle
center of ***Always
gravity
inside the
triangle
Centroid The centroid of
Theorem a triangle is
located of the
distance from
each vertex to
the opposite
side
Altitude of A perpendicular
a Triangle segment from a
vertex to the
line containing
the opposite
side.
***Can be
inside, outside,
or on the
triangle.
Orthocente The point
r of the
of
Triangle
concurrenc
y of the
altitudes of
a triangle
5-4 The Triangle Midsegment Theorem
Midsegment A segment
of a Triangle that joins the
midpoints of 2
sides of the
triangle
Triangle
A midsegment
Midsegment of a triangle is
Theorem
parallel to a
side of the
triangle, and
its length is
half the length
of the side.
Solving Compound Inequalities
5-5 Indirect Proof & Inequalities on One Triangle
Writing an
Indirect proof
1. Identify the conjecture to be
proven.
2. Assume the OPPOSITE (the
negation) of the conclusion is true.
3. Use direct reasoning to show that
the assumption has led to a
contradiction.
4. Conclude that since the
assumption is false, the original
conjeture must be true.
Angle Side
If 2 sides of a triangle
Relationships in are not congruent,
Triangles
then the larger angle
is opposite the longer
side.
If 2 angles of a
triangle are not
congruent, then the
longer side is
opposite the larger
angle.
Triangle
The sum of any 2
Inequality
sides of a triangle is
Theorem
greater than the
length of the 3rd side.
5-6 Inequalities in 2 Triangles
Inequalities in 2 Hinge Theorem
Triangles
If 2 sides of 1 triangle
are congruent to 2
sides of another
triangle and the
included angles are
not congruent, then
the longer side is
across from the
larger included angle.
Converse of the
Hinge Theorem
If 2 sides of a triangle
are congruent to 2
sides of another
triangle and the
included 3rd sides are
not congruent, then
the larger angle is
across from the
longer 3rd side.
Simplest Radical Form
5-7 The Pythagorean Theorem
The
Pythagorean
Theorem
The sum of the
squares of the legs
of a right triangle is
equal to the square
of the hypotenuse.
Converse of the If the sum of the
Pythagorean
squares of the legs
Theorem
of a triangle is equal
to the square of the
third side, then the
triangle is a right
triangle.
Pythagorean
In
, c is the
Inequalities
length of the
Theorem
longest side. If
then the
triangle is obtuse
In
, c is the
length of the
longest side. If
then the
triangle is acute
5-8 Applying Special Right Triangles
45-45-90
Triangle
Theorem
30-60-90
Triangle