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Delia Coloma 9-5
Perpendicular bisector theorem: This theorem says
that if there is a point on the perpendicular
bisector, then the distance will be the same from
the endpoints of the segment.
Converse of the p.b.t: The converse says that if the
distance is the same from the endpoints of the
segment then it is on the perpendicular bisector.
Angle bisector theorem: This theorem says that if a
point is on the bisector of an angle, then it will
have the same measure from the sides of the
angle.
Converse of A.B.T: if a point in the interior of the
angle has the same distance then it is on the
bisector of an angle.
CONVERSE OF THE ANGLE
BISECTOR THEOREM
Concurrency is when three or more lines intersect
at a point. There are many types of them.
Intersect at one point
In the circumcenter theorem you will see that the three
perpendicular bisectors of the triangle are concurrent.
The circumcenter theorem says that
the circumcenter of a triangle has
the same distance from the vertices
of the triangle.
Where perpendicular bisectors meet!
Right triangles:
Acute triangles:
Obtuse angles:
A median of a triangle is a segment on which
one endpoint is a vertex and the other is the
midpoint of the other/opposite side.
The incenter of a triangle has the same
distance from the sides to the triangle.
THE INCENTER WILL ALWAYS BE
INSIDE OF THE TRIANGLE.
Were the angle bisectors meet! It can
be useful when putting something in the
middle of highways.
Right triangle:
Acute tiangle:
Obtuse triangle:
Centroid: the point of concurrency of the three
medians of a triangle.
It balances it so its distance from the vertex to
the other is doubled.
Centroid theorem: states that the centroid of a
triangle is located 2/3 of the distance from a vertex
to the midpoint of the opposite side.
Can be helpful when
building something so
that it is balanced.
This means that it is the point on
which the medians intersect.
A perpendicular segment from the vertex to the opposite side line.
TRIANGLES HAVE 3 ALTITUDES
It can be inside, outside or on it.
The two lines containing the altitudes
are concurrent to the line
intersecting.
Concurrency of altitudes in a
triangle means/is the point where
the altitudes intersect.
A segment that joins the midpoints of two sides of a triangle.
TRIANGLES HAVE 3 AND THEY FORM A MIDSEGMENT TRIANGLE.
The midsegment of a triangle is parallel to the opposite side of the
triangle, and it is half its size.
2 in
4 in
10 cm
if two sides of a triangle are not congruent, then
the larger angle is opposite the longer side.
If two angles of a triangle are not congruent,
then the longer side is opposite the larger
angle.
This means that the non-adjacent
interior angles are smaller than the
exterior angle.
When you add them up you get the
measure of the exterior angle.
A
60
60
B
C
120
70
70
140
80
40
40
C
This theorem says that the sum of any two sides of a
triangle is greater that the other side.
X
A
WY<WY+YX
B
AC+CB > AB
U
TU+TV>UV
W
Y
T
6
V
C
CA+AB>CB
A
B
Indirect proofs are used when something is not
possible to be proved directly.
STEPS WHEN DOING AN IDIRECT PROOF:
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 its true.
Prove: a triangle cannot have 2 right triangles.
statements
reasons
Assume a triangle has 2
right triangles <1+<2
Given
m<1= m<2 = 90
Def. of Right triangle
m<1+ m<2 = 180
Substitution
m<1+ m<2 + m<3= 180
Triangle sum theorem
M<3=0
Contradiction
So it cannot have 2 right angles.
Write an indirect proof that the supplementary of an acute
angle cannot be an acute angle.
statements
reasons
Assuming that the supp. of an
acute < is an acute <.
Given
2 <´s added= 180
Def of supplementary
Acute < less than 90
Def of acute
Angle addition property
contradiction
So the supplement of an
acute angle cannot be an
acute <.
Prove: A triangle has two obtuse
angles.
Statements
reasons
Assume a triangle has 2
obtuse angles <1+<2
given
<1=<2= 91+
Def of obtuse angle
<1+<2=182+
substitution
A triangle´s angles only
measure up to 180
contradiction
A triangle cannot have 2
obtuse angles.
BC>EF
This theorem says that if two sides of a triangle
are congruent to two sides of another and the
included angles are not, then the long third
side is across from the larger included angle.
BC>DF
ED>AB
If two sides if a triangle are congruent to another two sides of
other triangle and the third sides are not congruent, then the
larger included angle is across from the third side.
M<EFD > M<ACB
M<BAC > M<EDF
m<BAC > m<DEF
It is called the Theorem 5-8-1 it states that in a 45-45-90
triangle both legs are congruent, the length of the
hypotenuse us the length of a leg times √2.
To do it just multiply a leg times √2. Your answer could be
in radical form.
Radical=
√
x
find the value of x
x
5
7
7x√2= 7√2
X= 7√2
5
5x √2= 5√2
X= 5√2
9√2
x
x
9√2 /√2 = 9
X= 9
7
It is called the theorem 5-8-2, it says that the length of
a hypotenuse is 2 times the length of the shorter leg.
The length of the longer leg is the length of the shorter
leg times √3.
18
Not drawn in scale
9√3
9
√3
12
6
6√3
7
7√3
14