Theorem
If
If a segment joins
the midpoints of
two sides of a
triangle, then the
segment is
parallel to the
third side and is
half the distance.
D is the midpoint
of CA and
E is the midpoint
of CB
A
Then
DE // AB and
1
DE  AB
2
C
D
C
E
D
B
A
x
2x
E
B
 Find
all missing side lengths, given AB = 10,
BE = 7, and DE = 4
AD = 5
AC = 8
DB = 5
EC = 7
EF = 5
DF = 7
AF = 4
BC = 14
A
10
D
FC = 4
F
4
B
E
7
C

Solve for x if the perimeter of triangle ABC is 42
A
12  9  (3x  3)  (3 x  3)  42
15  6x  42
12
3x – 3
D
6x  27
9
F
x  4.5
B
3x - 3
E
C
3x - 3
 P.
288 #’s 10-18 even, 19-27, 29-32, 37, 4044, 49-52
Theorem
If a point is on
the perpendicular
bisector of a
segment, then it
is equidistant
from the
endpoints of the
segment.
If
Then
PM  AB and
MA  MB
PA  PB
P
A
M
P
B
A
M
B
Theorem
If a point is
equidistant from
the endpoints of
the segment,
then it is on the
perpendicular
bisector of the
segment.
If
Then
PM  AB and
MA  MB
PA  PB
P
A
P
B
A
M
B
 Calculate
AC
4x  6x 10
A
2x  10
x5
x5
4x
6x – 10
AC  6(5)  10
AC  30 10
AC  20
B
D
C
Complete Got it 1, p. 293
Ans: QR = 8

The distance from a point to a line is the length
of the perpendicular segment from the point to
the line.
A
l
B
Theorem
If
Then
If a point is on
the bisector of an
angle, then the
point is
equidistant from
the sides of the
angle.
QS bisects PQR,
SP  SR
SP  QP, and
SR  QR
P
P
S
Q
S
Q
R
R
Theorem
If a point in the
interior of an
angle is
equidistant from
the sides of the
angle, then the
point is on the
angle bisector.
If
Then
QS bisects PQR
SP  QP,
SR  QR, and
SP  SR
P
P
S
Q
S
Q
R
R
 Calculate
RM
7 x  2x  25
5x  25
R
7x
M
2x+25
A
P
x5
RM  7(5)
RM  35
Complete Got it 3, p. 295
Ans: FB = 21
 P.
296 #’s 6-8, 12-22, 36, 39-46
Theorem
The
perpendicular
bisectors of the
sides of a triangle
are concurrent at
a point
equidistant from
the vertices
B
Diagram
Symbols
Perpendicular
bisectors
A
PX , PY , and PZ
are concurrent at P.
X
Y
PA  PB  PC
P
Z
C

When 3 or more lines intersect at one point,
they are concurrent. The point at which they
intersect is the point of concurrency.

The point of concurrency of the perpendicular
bisectors of a triangle is called the circumcenter
of the triangle.

Since the circumcenter is equidistant from the
vertices, you can use the circumcenter as the
center of the circle that contains each vertex of
the triangle. You say the circle is circumscribed
about the triangle.
A
X
Y
P
B
Z
C
 Find
the circumcenter of the following 3
points.
Circumcenter
Theorem
The bisectors of
the angles of a
triangle are
concurrent at a
point equidistant
from the sides of
the triangle.
Diagram
Symbols
Angle bisectors
AP, BP, and CP
A
are concurrent at P.
PX  PY  PZ
X
Y
P
B
Z
C
The point of concurrency of the angle bisectors
of a triangle is called the incenter of the
triangle.
 P is the center of the circle that is inscribed in
the triangle.
A

X
Y
P
B
Z
C
 If
P is the incenter and PY = 2x-4 and PZ =
5x-13, find BP, if PB = 3x.
2 x  4  5 x  13
9  3x
A
PB  3(3)
X
3x
3 x
Y
P
PB  9
2x-4
5x-13
B
Z
C
 P.
305 #’s 8-12 even, 15-18, 22, 33-34, 37-40

A median of a triangle is a segment whose
endpoints are a vertex and the midpoint of the
opposite side.
D
Median
E
F
J
Theorem
Diagram
The medians of a
triangle are concurrent
at a point that is two
thirds the distance
from each vertex to
the midpoint of the
G
opposite side.
Symbols
2
DC  DJ
3
2
EC  EG
3
2
FC  FH
3
D
H
C
E
F
J

In a triangle, the point of concurrency of the
medians is the centroid of the triangle. The
point is also called the center of gravity of a
triangle because it is the point where a
triangular shape will balance. For any triangle,
the centriod is always inside the triangle.
 In
the diagram below CE = 20, CH = 6, and DC
= 8. Find CG, FC and DJ.
2
CE  GE
3
D
G
10
F
8
C
6
H
20
J
2
20  GE
3
3
2
3
20

GE
  
E

2
3

2
GE  30
CG  30  20  10
 In
the diagram below CE = 20, CH = 6, and DC
= 8. Find CG, FC and DJ.
1
CH  FH
3
1
6  FH
3
D
G
10
12
F
8
C
6
H
20
J
E
3
1
3
 6    FH 
1
3
1
FH  18
FC  18  6  12
 In
the diagram below CE = 20, CH = 6, and DC
= 8. Find CG, FC and DJ.
2
DC  DJ
3
2
8  DJ
3
D
G
10
12
F
8
C
6
H
20
J
Complete Got it 1, p. 310
E
3
2
3
 8    DJ 
2
3
2
DJ  12
Ans: a) ZC = 13.5
b) 2 : 1

An altitude of a triangle is the perpendicular
segment from the vertex of the triangle to the
line containing the opposite side. An altitude of
a triangle can be inside or outside the triangle,
or it can be a side of the triangle.
D
D
D
E
F
E
E
F
F
Theorem
Diagram
D
The lines that
contain the
altitudes of a
triangle are
concurrent.
P
E
F

The lines that contain the altitudes of a triangle
are concurrent at the orthocenter of the
triangle. The orthocenter can be inside, on, or
outside the triangle.
D
D
E
F
D
E
E
F
Acute
Triangle
F
Obtuse
Triangle
Right
Triangle

Triangle ABC has vertices A (1,3), B (2,7), and C
(6,3). What are the coordinates of the
orthocenter of the triangle?
Graph the points
Create two altitudes
Locate the point of
intersection
The orthocenter is the
point (2, 4)
Complete Got it 3, p. 311
Ans: (1, 2)
 P.
312 #’s 8-20, 24-27, 31, 37-41, 43-44
Comparison Property of Inequality
If a  b  c and c  0, then a  b
b
c
a
Corollary
If
Then
The measure of
an exterior angle
of a triangle is
greater than the
measure of each
of the remote
interior angles
1 is an exterior angle
2
3
1
m1  m2 and
m1  m3
Theorem 5-10
If
If two sides of a
triangle are not
congruent, then
the larger angle
lies opposite the
longer side.
mY  mZ
XZ  XY
Y
Z
X
Theorem 5-11
If two angles of a
triangle are not
congruent, then
the longer side
lies opposite the
larger angle.
Then
If
Then
mA  mB
C
B
A
BC  AC
 In
the diagram, list the sides in order from
smallest to largest.
C
110°
B
AC , BC , BA
30°
mA  180  (30  110)
mA  40
Complete Got it 3, p. 327
A
Ans: OX
Theorem
Diagram
The sum of
the lengths of
any two sides
of a triangle is
greater than
the third side.
Y
X
XY  YZ  XZ
YZ  XZ  XY
XZ  XY  YZ
Z
 Can
a triangle have side lengths of 4, 4, and
8? How about 4, 5, and 6?
44 8
45  6
46  5
65  4
88
96
10  5
11  4
No, a triangle
cannot be made
with side lengths
of 4, 4, and 8.
Yes, a triangle can be made with
side lengths of 4, 5, and 6. The sum
of the lengths of any two sides is
greater than the third.
Complete Got it 4, p. 327
Ans: a) No
b) Yes

If two sides of a triangle are 8 and 14, what is
the range of possible lengths for the third side?
8 14  x
8  x  14
22  x
x 14  8
x6
x  6
The length of a side of a triangle can never be negative so
we can eliminate that answer. So the range of values is
between 6 and 22.
6  x  22
Complete Got it 5, p. 328
Ans: 3 < X < 11
 P.
328 #’s 6-34 even, 37-39, 43, 44, 46-49,
53-55.
Theorem 5-13
If two sides of one
triangle are
congruent to two
sides of another
triangle, and the
included angles
are not congruent,
then the longer
third leg is
opposite the
larger included
angle.
If
Then
mA  mX
BC  YZ
B
A
C
Y
X
Z
Theorem 5-14
If two sides of one
triangle are
congruent to two
sides of another
triangle, and the
third sides are not
congruent, then
the larger
included angle is
opposite the
longer third side.
If
Then
mA  mX
BC  YZ
B
A
C
Y
X
Z
 What
is the range of possible values for x?
Find the upper limit
R
15
S
10
60°
U
(5x-20)°
T
Complete Got it 3, p. 335
60  5x  20
80  5x
16  x
Find the lower limit
5x  20  0
5x  20
x4
4  x  16
Ans: -6 < x < 24
 P.
336 #’s 6-9, 11-14, 16-18, 21, 24, 26-28,
31-35, 37-39