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Chapter 5
Relationships in Triangles
5.1 Bisectors, Medians and
Altitudes
Special Segments of
Triangles
Median – A segment drawn from the vertex
of a triangle to the midpoint of the opposite
side.
Altitude – A segment drawn from the vertex
of a triangle perpendicular to the line that
contains the opposite side.
Angle Bisector – A segment drawn from the
vertex of a triangle to the opposite side,
dividing the vertex angle into two congruent
angles.
Special Segments (Con’t)
Perpendicular Bisector – A segment that
starts at the midpoint of a side and is
perpendicular to that side.
Notice the similarities and differences
among the four different segments.
Medians
C
D
A
G
E
F
To draw a median, you start at a vertex
and end at the midpoint of the opposite
side.
What is point D?
B
Since there are three vertices
in a triangle, then there are
three medians of a triangle.
The intersection of the three Medians is called the Centroid.
Centroid Theorem – The Centroid is located 2/3rds the way
from the vertex to the midpoint on the opposite side.
Angle Bisectors
C
To draw an <bis, you start at a vertex
and end at the opposite side.
5 6
D
G
D is NOT the midpoint of segment AC.
1 2
2
1
3
4
A
F
E
B
3
4 and
5
The intersection of the three <bis is called the Incenter.
Incenter Theorem – The Incenter is equidistant from the
three sides of the triangle.
Any point on an <bis is equidistant from the two sides of the
angle bisected.
6
Altitudes
C
To draw an altitude, you start at a vertex
and perpendicular to the line that
contains the opposite side.
D is NOT the midpoint of segment AB.
A
D
B
The intersection of the three altitudes is called the Orthocenter.
Fact: Regardless of what type of triangle you have, the
Incenter and Centroid are always in the interior of the triangle.
The Orthocenter is interior when the triangle is acute, exterior
when the triangle is obtuse and at the right angle when the
triangle is a right triangle.
Perpendicular Bisectors
C
To draw a perpendicular bisector you start
at the midpoint of a side and draw a
segment perpendicular at that point.
The intersection of the three
perpendicular bisectors is called
the Circumcenter.
A
B
The circumcenter is equidistant from the three points of the
triangle.
Any point on the perpendicular bisector is equidistant
from the end points of the segment bisected.
The location of the circumcenter is also dependant on the
classification of the triangle, Acute – Interior, Obtuse – Exterior
Right – Midpoint of the Hypotenuse.
5.2 Inequalities and Triangles
Properties of Inequalities
Comparison Property –
a<b, a=b or a>b
Transitive Property –
If a>b and b>c, then a>c
If a<b and b<c, then a<c
Addition/Subtraction Property –
If a<b, then a + c < b + c.
Multiplication/Division Property –
If a<b, then ac < bc (unless c is negative)
Exterior Angles
Exterior Angle Theorem –
3
The measure of the exterior is
equal to the sum of the two
1
2
remote interior angles.
(m<1 = m<2 + m<3)
Since the measurement of <1 is equal to the sum
of the two remote interior angles, then the
measurement of angle 1 must be greater than
either remote interior angles.
(m<1 > m<2,
m<1 > m<3)
Relationships
There is a unique relationship between the
sides and the angles that are opposite of the
sides.
If one side of a triangle is larger than another
side, then the angle opposite the larger side is
larger than the angle opposite the shorter side.
If one angle of a triangle is larger than the other
angle, then the side opposite the larger angle is
larger than the angle opposite the smaller
angle.
Example
C
13
12
A
20
Knowing the measurements
of the three sides we can list
them from largest to smallest.
B
Because the relationship
of the sides is the same
AB, BC, and AC
as the relationship of the
m<C, m<A, and m<B angles opposite them, we
can list the angles
opposite them from
m<C > m<A > m<B
largest to smallest.
Another Example
C
Knowing the measurements
of the three angles we can list
them from largest to smallest.
100°
55°
A
25°
B
Because the relationship
m<C, m<A, and m<B of the angles is the same
as the relationship of the
AB, BC, and AC
sides opposite them, we
can list the sides opposite
them from Largest to
AB > BC > AC
Smallest.
Another Example (H)
B
60°
70°
II
60°
A
50°
65°
I
55°
D Given the measurements of
the angles find the largest
and smallest segment in this
picture.
C
I - <D, <B and <C
So; BC > CD > BD
II - <B, <A and <C
So; AC > BC > AB
Since BC is in common between the two triangles….
We can conclude that AC is the largest segment and
BD is the smallest?
5.3 Indirect Proof
Honor’s Only
Indirect Proof
Another reasoning we can use is Indirect
Reasoning – that is prove that your
statement can’t be true.
To do an Indirect Proof – you first negate
the conclusion.
Then you make that your Hypothesis,
Then prove that the conclusion can’t be
reached.
Bottom Line
Given P, prove Q via Indirect Proof.
(P → Q)
First, write contra positive (~Q → ~P)
Where ~Q is the given and you want to
prove ~P is not possible.
You will not have to do one, just know how
to set one up.
5.4 Triangle Inequality
Triangle Inequality
Triangle Inequality Theorem – In a
triangle, the sum of the measurements any
two sides must be greater than the
measurement of the third side.
B
A
AB + BC > AC
BC + AC > AB
AC + AB > BC
C
Two Types of Problems
There are two types of problems that deal
with the Triangle Inequality Theorem
1st – Given three sides, can you make a
triangle?
2nd – Given two sides, what is the range of
values possible for the third side
1st Type of Problem
Given three sides, can you make a
triangle?
Given segments that measure 3”, 5” and
10”, can you make a triangle?
Short Cut – look at two smallest
measurements, is their sum greater than
the third?
No, then no triangle, if yes, then there is a
triangle.
2nd Type Problem
Given two sides, what is the range of
values possible for the third side.
The range is found by finding the sum and
difference between the two given
measurements.
Ex: Find the range of the 3rd side if two
sides measure 12 and 15.
The range of the 3rd side is between 3 (15
– 12) and 27 (15 + 12).
5.5 Inequalities Involving 2
Triangles
Background
Everything we’ve done up to this point
deals with inequalities in one triangle.
Triangle Inequality Theorem – Sum of any
two sides is greater than the 3rd side.
Opposite Angle Theorem – Says that the
largest angle is always opposite the
largest side.
Opposite Side Theorem – Says that the
largest side is always opposite the largest
angle.
Inequalities in 2 Δ’s
There are two theorems that deals with
inequalities in 2 triangles.
SSS Inequality –
SAS Inequality –
SSS Inequality
SSS Inequality (Hinge Theorem) – States
that if two sides of one triangle are
congruent to two sides of another triangle,
then the relationship of the third set of
sides governs the relationship of the
included angles.
F
B
35
30
A
C
D
Since BC < FE then m<A < m<D.
E
SAS Inequality
SAS Inequality – States that if two sides of
one triangle are congruent to two sides of
another triangle, then the relationship of
the included angles governs the
relationship of the sides opposite those
included angles.
F
B
30°
A
35°
C
D
Since m<A < m<D then BC < FE
E