# Properties of Triangles

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```					     Chapter 5:
Properties of Triangles
Geometry Fall 2008
5.1 Perpendiculars and Bisectors

   Perpendicular Bisector: a segment, ray, or line
that is perpendicular to a segment at its
midpoint.

   Equidistant: the same distance from
Perpendicular Bisector Theorem
      Bisector Theorem: If a point is on the
bisector of a segment, then it is equidistant
from the endpoints of the segment.

P                  If PM bisects
AB and is to
AB, then
A            M           B      AP = BP.
Converse

   If AP = BP, then P is on the   bisector of AB.
Angle Bisector Theorem
   If a point is on the bisector of an angle then it
is equidistant from the two sides of an angle.
B

D
If m BAD=m CAD
A                               then DB = DC.
C

Converse: If DB = DC then m BAD = m CAD
5.2 Bisectors of a Triangle
   Perpendicular bisector of a triangle: line, ray,
or segment that is perpendicular to a side of
the triangle at the midpoint of the side.
Concurrent Lines
   When 3 or more lines intersect in the same
point they are called concurrent lines.

   Then intersection is called the concurrency.

   The 3 perpendicular bisectors of a triangle are
concurrent.
Acute, Right and Obtuse

   Inside        On        Outside
Circumcenter
   The intersection of the three perpendicular
bisectors is called the circumcenter.

   The circumcenter is equidistant from each of
the vertices.
Angle Bisector of a Triangle
   Angle bisector of a triangle: a line, segment, or
ray that bisects and angle of the triangle
Incenter

   The 3 angle bisectors are concurrent.

   The intersection of the 3 angle bisectors is
called the incenter.
Incenter
   The incenter is always inside the triangle and
is equidistant from the sides of the triangle.
   GB = GD = GF
D           F

G

B
Is C on the perpendicular
bisector of AB?

A                     B

C
Is C on the perpendicular
bisector of AB?

C

A            B
Is P on the bisector of   A?

P
A
Is P on the bisector of A?

4

A             P

4
5.2 examples
B
Find the length of DA.

Find the length of AB.

E             D   Why is ADF =      BDE?

2

A
C         F
Examples
   V is the incenter of         XWZ.
   Length of VS:
   m VZX =
W

S           T
3

V
20

Z
X                           Y
Find ID
C

8

10
D

I

A                    B
Find BD
C

D

15

A
B   12             12
Homework

p. 268 # 8-13, 16-26
p. 275 # 10-17
5.3 Medians and Altitudes of a
Triangle
   Median of a Triangle: a segment whose
endpoints are a vertex of the triangle and the
midpoint of the opposite side.
Centroid
   The three medians are concurrent.
   The intersection of the three medians is called
the centroid of the triangle.
Concurrency of Medians of a
triangle
   The medians of a triangle intersect at a point
that is two thirds of the distance from each
vertex to the midpoint of the opposite side.
B
 If P is the centroid
of ABC, then                     D

AP = 2/3 AD,
C
BP = 2/3 BF,         E
P
and CP = 2/3 CE.                 F

A
Altitudes of a Triangle
   The altitude of a triangle is the perpendicular
segment from a vertex to the opposite side or
to the line that contains the opposite side.

   The intersection is called the orthocenter.
BAE = EAC and BF = FC
   Median:
   Altitude:         A

     Bisector:
     Bisector:
   a) AD
   b) AE                 G

   c) AF
B   D   E   F   C
   d) GF
C is the centroid of XYZ. YI = 9.6
X
   CK =
   XK =
   YC =                         8
J                   I
   KZ =
C
   JZ =
   Perimeter of :
Y               K           Z
5
5.4 Midsegment Theorem
   A midsegment of a triangle is a segment that
connects the midpoints of two sides of a
triangle.
Midsegment Theorem
   The segment connecting the midpoints of two
sides of a triangle is parallel to the third side
and is half as long.
Example 1
   D, E, and F are the midpoints of the sides.
A
   DE ll _____
   FE ll _____
   If AB = 14,             D               F

then EF = _____
   If DE = 6,
B
C
then AC = _____                  E
Example 2
   R, S, and T are the midpoints of the sides.
   RK = 3, KS = 4, and JK KL
J
   RS = _____
   JK = _____
   RT = _____          R
T

   Perimeter = _____ 3

K     4      S           L
Example 3
   If YZ = 3x + 1, and MN = 10x – 6,
then YZ = ______
M

X
Z

O               Y                 N
Example 4
   If m MON = 48 , then m MZX = _____

M

X
Z

O              Y             N
Example 5
   Find the midpoint of JK.

   Find the length of NK.

   Find the length of NP.

   Find the coordinates of
point P.
Homework for 5.3-5.4

   P. 282 #8-11, 17-20
   P. 290 # 12-18, 28-29
5.5 Inequalities in One Triangle
   If side 1 of a triangle is larger than side 2, then
the angle opposite side 1 is larger than the
angle opposite side 2.
B

Since BC > AB, A > C
5
3

A                             C
Converse
   If angle 1 is larger than angle 2, then the side
opposite angle 1 is longer than the side
opposite angle 2.
B

Since A > C, BC > AB

60                 40
A                            C
Exterior Angle Inequality
   The measure of an exterior angle of a triangle
is greater than the measure of either of the two
nonadjacent interior angles.
A

1
B
C

   m 1 > m A and m 1 > m B
Triangle Inequality
   The sum of the lengths of any 2 sides of a
triangle is greater than the length of the 3rd side
   AB + BC > AC                                  A

   AC + BC > AB
   AB + AC > BC

C                B
Example 1
   Name the shortest and longest sides.
Q

65                       35
P                                  R
Example 2
   Name the smallest and largest angles.
Z

4

X

6
8

Y
Example 3
   List the sides in order from shortest to longest.

B
C
95           30

A
Example 4
   Find the possible measures for XY in   XYZ.

   XZ = 2 and YZ = 3
Example 5
   Determine whether the given lengths could
represent the lengths of the sides of a triangle:

   A) { 5, 5, 8}

   B) { 5, 6, 7}

   C) { 1, 2, 5}

   D) { 7, 3, 9}
Homework 5.5

   Pg. 298 # 6-11, 14-19, 24-25
5.6 Indirect Proof and
Inequalities in Two Triangles

   Indirect Proof: a proof in which you prove that
a statement is true by first assuming that its
opposite is true.
Writing an indirect proof
   Assume that the negation of the conclusion
(what you are trying to prove) is true.

   Show that the assumption leads to a
contradiction of known facts or of the given
information.

   Conclude that since the assumption is false the
original conclusion is true.
Example 1
   Given: ABC
   Prove: ABC does not have more than one
obtuse angle.

   Begin by assuming that ABC does have more
than one obtuse angle.
   m A > 90 and m B > 90
   m A + m B > 180
Example 1
   You know that the sum of the measures of all
three angles is 180 .
   m A + m B + m C = 180
   m A + m B = 180 - m C
   You can substitute 180 - m C for
m A + m B in m A + m B > 180 .
   180 - m C > 180
   0 >m C
Example 1
   The last statement is “not possible.” Angle
measure in triangles cannot be negative.

   So you can conclude that the original
assumption must be false. That is, ABC
cannot have more than one obtuse angle.
Hinge Theorem
   If 2 sides of one triangle are congruent to two
sides of another triangle, and the included
angle of the first is larger than the included
angle of the second, then the third side of the
first is longer than the third side of the second.
R                                 V

100                 80
S               T                     X
W
Converse of Hinge Theorem
   If two sides of one triangle are congruent to
two sides of another triangle, and the third side
of the first is longer than the third side of the
second, then the included angle of the first is
larger than the included angle of the second.
R                              V

10                       8

S              T                    X
W
Example 1
   Complete with <, >, or =.
   AB ____ DE
C

105
B

E

A
D
110
F
Example 2
   Complete with <, >, or =.
   m 1 ____ m 2

2

8
7

1
Homework 5.6
   Pg. 305 # 3-5, 6-20

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