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Chapter 5 Journal
Definition of Perpendicular bisector: a line perpendicular
to a segment at the line segment´s midpoint
Perpendicular bisector theorem: If a line is
perpendicular, then it is equidistant from the endpoints
of a segment.
Converse of perpendicular bisector theorem: If a point is
equidistant from the endpoint of a segment, then it is
perpendicular line.
Perpendicular bisector
theorem
1.
2.
3.
=
=
=
Converse of perpendicular
bisector theorem
Angle bisector theorem: a ray or line that
cuts an angle into 2 congruent angles. It
always lies on the inside of an angle
Converse of angle bisector theorem: If a
point is equidistant from the sides of a
angle, then it lies on the bisector angle.
Angle bisector theorem
A
B
C
C
B
A
A
C
B
AB = CB
A
D
B
Converse of
angle
bisector
theorem
C
C
A
D
A
D
B
B
<ADB = <CDB
(Congruent ,not equal)
C
Definition of concurrency: Where three or
more lines intersect at one point.
Concurrency of perpendicular bisectors: Point where
the perpendicular bisectors intersect.
Definition of Circumcenter: the point of
congruency where the perpendicular
bisectors of a triangle meet.
The circumcenter theorem: The
circumcenter of a triangle is equidistant
from the vertices of the triangle.
Concurrency of angle bisectors: Point where the
angle bisectors intersect.
Definition of incenter: The point where
the angle bisector intersect of a
triangle
Always occur on the side of triangle
Incenter theorem: The incenter of a
triangle is equidistant from the side of
a triangle
Definition of
Median: segment
that goes from the
vertex of a triangle
to the opposite
midpoint.
Centroid: The point where the
medians of a triangle intersect.
The distance from the vertex to
the centroid is double the
distance from the centroid to the
opposite midpoint.
Concurrency of medians: point where the medians
intersect.
Definition of altitude: a
segment that goes from the
vertex perpendicular to the
line containing the opposite
side.
Definition of Orthocenter: Where
the altitudes intersect
If the triangle is acute, the
orthocenter is on the inside of the
triangle
If it is right orthocenter is on the
vertex of the right angle.
Concurrency of altitudes: point where the altitudes
intersect.
Midsegment of a triangle:
segment that joins the midpoints
of two sides of the triangle
A midsegment of a triangle, and
its length is half the length of that
side.
Triangle midsegment theorem: A midsegment of a triangle
is parallel to a side of the triangle, and its length is half the
length of that side.
Hinge theorem: If 2 triangles have 2 sides that are
congruent, but the third side is not congruent, then
the triangle with the larger included angle has the
longer third side.
Converse of Hinge theorem: If two sides of a triangle
are congruent to the two sides of the other triangle
but the other sides are not congruent then, the
largest included angle is across from the largest
side.
B
Hinge Theorem
H
J
A
<B > <Y
AC > XZ
J>A
A
B
C
I
H
J
C
I
Converse of Hinge Theorem
F
C
D
E
D
A
B
FE > CB,
FD=CA,
DE = AB
(congruent)
C
F
C
A
E
A
F
B
<D> <A
D
EE
B
Triangle side-angle relationship
theorem: In any triangle, the longest
side is always opposite from the
largest angle and vice versa.
Longest side
Shortest side
Longest side
The non-adjacent interior angles are smaller than the
exterior angle
A+B = exterior angle (c)
B
A
B
C
A
C
C
B
A
Triangle inequality theorem:
the 2 smaller sides of a
triangle must add up to
more than the length of the
3rd side.
4, 7, 10
4+7=11
YES
3, 1.1, 1.7
1.1+1.7= 2.8
NO
2, 9, 12
2+9=11
NO
Indirect proof: used when it is not possible to
prove something directly.
Steps:
1. Assume that what you are proving is false
2. Use that as your given, and start proving it
3. When you come to a contradiction you have
proved that it is true.
Prove: A triangle cannot have 2
right angles
A triangle has 2 right angles
(<1 & <2)
Given
M<1=m<2=90
Def. right angle
M<1+m<2=180
Substitution
M<1+m<2+m<3=180
Triangle sum theorem
M<3=0
contradiction
Proove: a right triangle cannot have an
obtuse angle
A right triangle can have an
obtuse angle (<A)
Given
M<A + m<B= 90
Substitution
M<A =90 – m<b
Subtraction prop.
M<A> 90°
Def. obtuse triangle
90° - m<b > 90
substitution
m<b = 0
contradiction
A triangle cannot have 4 sides
A triangle can have 4 sides
Given
A square is a shape with 4
sides
Def of square
A triangle is a shape with only Def of triangle
3 sides
A triangle cannot have 4
sides
contradiction
45° - 45° - 90° triangle theorem: In this kind of
triangle, both legs are congruent and the
hypotenuse is the length of a leg times √2
30° - 60° - 90° Triangle theorem: In this kind of
triangle the longest leg is √3 the shorter leg and
the hypotenuse is √2 the shortest side of the
triangle.
45° - 45° - 90° triangle theorem
A
BC=AC=X
AB=X√2
X
B
X
C
45° - 45° - 90° triangle theorem
X=14√2
45°
14
X
45°
30° - 60° - 90° triangle theorem
16
B
20=2x
10=x
Y=a√3
Y=10√3
16=2a
8=a
B=a√3
B=8√3
20
Y
100
d
100=2d
50=d
H=d√3
H=50√3