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What is a Perpendicular Bisector? A Perpendicular Bisector a line or ray
that passes through the midpoint of the segment
Theorem: If a line pass through a point in the segment that are
equidistant from the endpoint then is a perpendicular bisector.
Converse: If the line that passes though a point in the segment and its
not equidistant then is not a perpendicular bisector.
1) If AB=25.4cm AD=20cm and
DB=12cm find AB.
AB= 25.4
2) If H is perpendicular bisector of
CB and DB is 4. Find CB.
CB= 8
3) Given that H is perpendicular
bisector of segment CB and AB is
4x+4 and AC is 8x-6. Find AB
AB=AC
4x+6=8x-6
-4x
-4x
+6=4x-6
+6
+6
4x=12
X=3 AB=4(3)+4
AB=16
What is an Angle Bisector? A Ray or line that divides the angle is two
congruent angle
Theorem: If a point is in the line of the angle bisector then is
equidistant from the sides of the angles
Converse: If its equidistant from the sides of the angles then it on the
angle bisector
B
1) <ABD congruent <ACD
2) <DAC congruent <DAB
3) <BDA congruent <CDA
D
A
C
Counter:
1) Since AB is congruent to AB, AD is an angle bisector of triangle BAC
2) Since BD is congruent to DC, AD is an angle bisector of triangle BAC
What is concurrent: Concurrent is when three or more lines meet
together making a point.
The Perpendicular bisector of a triangle is a point where its
equidistant from the vertices of the triangle
Circumcenter is where the point of the 3
perpendicular bisector meets, its special
because the point is equidistant through all the
vertices of the triangle.
When a point is made and is equidistant from the sides of the triangles.
A point made by the meeting of the three angle
bisector, and it equidistant from the sides of the
triangle.
INCENTER
The median of a triangle is a segment that goes
from the vertex to the opposite side
midsegment
A point where all the median meets. The distant
from the centroid to the vertex will the double
from the distant from the centroid to the side.
The median of a triangle that
intersect a point creating a line
that is double the other part of
the line
2
1.75
3.3
2.5
2
3
5
6.6
4
1
3.5
0.4
1.43
6.2
3.1
0.2
2.86
6
Segment that is perpendicular to the other angles.
The point that all altitude lines meet of a
triangle.
A point that the
altitudes
intersects in a
triangle.
What is Midsegment? Is a segment joined by the midpoint of
the sides, the triangle has 3 midsegment in total.
Theorem: The midsegment is parallel to the opposite side and
it’s half the size of the side.
*ATTENTION* This Images are not draw in scale :P
EXAMPLES
In the triangle they are three segments that are related to the angle,
meaning that if one segment is the biggest then to the opposite side angle
will be the biggest one and if the segment is the shortest one the opposite
side angle will be the shortest one.
The angle that is an extension of a interior angle. A+B= the exterior angle
b
c
a
A+B
b
b
a
c
A+B
a
c A+B
1. 2,5,8  No, because adding 2+5<8 and this will not
make a triangle.
2. 15,28,40  Yes because adding 15+28>40 and this
will make a triangle
3. 16, 24, 40  No because adding 16+24=40 making
two line with the same lengths.
The sum of 2 sides will be greater than the 3rd side.
It’s a way to proof not directly but indirectly.
STEPS TO SOLVE THIS
1)Assume the Given is false
2) Use that as a give and start the proof
3)When you come to a contradiction that will be your last
step.
Prove A triangle cannot have 2 right angles
Statement
Reason
Assume a triangle has 2
right
<1=<2
Given
M<1=m<2=90 degrees
Def of Right Angle
M<1+M<2=180 degrees
Substitution
M<1+m<2+m<3=180
Triangle sum thrm
M<3=0
contradiction
A triangle cannot have 2 Right Angles
Proof: two supplementary angles cannot both be obtuse angles
Statement
Reason
Two supplementary
angles can both be
obtuse angles
Given
X>90 degrees
Def of obtuse angles
(X>90)+(x>90)
Substitution
(x>90)+(x>90)=180
contradiction
Two supplementary angles cannot both be obtuse.
A Triangle cannot have 2 obtuse angles
Statement
Reason
A triangle can have 2
obtuse angles
Given
X>90
Def of obtuse angles
M<1+m<2+m<3=180
Def of a triangle
X can be 91 or more
Def of obtuse angle
100+100+m<3=180
Substitution
200+M<3=180
Addition
M<3=-20
Subtraction
M<3=-20
contradiction
Triangle cannot have 2 obtuse angles
Two triangles with two congruent segments but the 3rd segment will
different and will be larger if the angle that is in the opposite side will
be bigger.
45-45-90: In this triangle the legs will the same size but the hypotenuse will
be length times √2
30-60-90: In this triangle the base will the smallest size the longest side would be
the hypotenuse that is the double of the base angle and the height
will be the base time the √3
45
X would be 1√2 because
you multiply the legs time √2
x
1
45
1
X would be 8√2 because we need
to divide 16 by √2, and we know
that we can’t have the √2 as a
denominator so we multiply
everything by √2 and later we have
16 √2 divided by 2 and we can
simplify by 2 and the answer would
be 8 √2
45
16
X
45
45
x
7
X would be 7 √2/2
30
45
x
y
X is going to be 8
Y will be 4 √3
60
4
30
X= 9 because the
hypotenuse is the double of
the base of the triangle
Y= is going to be 9 √3
because we already know
that the base is 9
18
y
60
x
30
y
X=3 √3
Y=6 √3
because to find the shortest
side we need to dived 9 by √3
that is 3 √3 making X and we
multiply time 2 the 3 √3
x
60
9