Transcript Angle Bisector Theorem

Triangles and their properties
•Triangle Angle sum Theorem
•External Angle property
•Inequalities within a triangle
•Triangle inequality theorem
•Medians
•Altitude
•Perpendicular Bisector
•Angle Bisector
Triangle Angle Sum Theorem
• The sum of the measures of the angles of a
C
triangle is 180°.
m∠A + m∠B + m∠C = 180
Ex: If m∠A = 30 and m∠B = 70;
what is m∠C ?
B
A
m∠A + m∠B + m∠C = 180
30 + 70 + m∠C = 180
100 + m∠C = 180
m∠C = 180 – 100 = 80
2
Exterior Angle Theorem
P
The measure of an exterior angle of a triangle is equal to sum
of its ___________________
remote interior angles
In the triangle below, recall that 1, 2, and
3 are interior
_______ angles of ΔPQR.
1
2
Q
3 4
R
Angle 4 is called an exterior
_______ angle of ΔPQR.
An exterior angle of a triangle is an angle that
forms a linear
_________,
pair (they add up to 180) with one of the angles of the triangle.
In ΔPQR, 4 is an exterior angle because 3 + 4 = 180.
Remote interior angles of a triangle are the two angles that do not form
____________________
a linear pair with the exterior angle.
In ΔPQR, 1, and 2 are the remote interior angles with respect to 4.
Exterior Angle Theorem
1
In the figure, which angle is
the exterior angle? 5
which angles are the remote
the interior angles? 2 and 3
If 2 = 20 and 3 = 65 , find 5
2
20 

65
3 
60
85 
If 5 = 90 and 3 = 60 , find 2 30 
4
90
5 
Exterior Angle Theorem
Exterior Angle Theorem
1 and 3
Inequalities Within a Triangle
If the measures of three sides of a triangle are unequal,
then the measures of the angles opposite those sides
are unequal ________________.
in the same order
P
11
M
8
13
L
LP < PM < ML
mM < mL < mP
Inequalities Within a Triangle
If the measures of three angles of a triangle are unequal,
then the measures of the sides opposite those angles
are unequal ________________.
in the same order
W
45°
75°
60°
J
mW < mJ < mK
JK < KW < WJ
K
Inequalities Within a Triangle
In a right triangle, the hypotenuse is the side with the
greatest measure
________________.
W
5
3
X
4
WY >
XW
WY >
XY
Y
Inequalities Within a Triangle
The longest side is
BC
So, the largest angle is
The largest angle is
A
L
So, the longest side is
MN
Triangle Inequality Theorem
The sum of the measures of any two sides of a triangle is
greater than the measure of the third side.
_______
b
Triangle
Inequality
Theorem
a+b>c
a
a+c>b
c
b+c>a
Triangle Inequality Theorem
Can 16, 10, and 5 be the measures of the sides of a triangle?
No!
16 + 10 > 5
16 + 5 > 10
However, 10 + 5 > 16
Medians, Altitudes,
Angle Bisectors
Perpendicular Bisectors
Every triangle has
1. 3 medians,
2. 3 angle bisectors and
3. 3 altitudes.
Just to make sure we are
clear about what an opposite
side is…..
B
A
C
Given ABC, identify the opposite side
1. of A.
BC
2. of B.
AC
3. of C.
AB
A new term…
Point of concurrency
• Where 3 or more lines
intersect
Definition of a Median of a Triangle
A median of a triangle is a segment whose
endpoints are a vertex and a midpoint of the
B
opposite side
Any triangle has three
medians.
L
M
A
N
Let L, M and N be the midpoints of AB, BC and AC respectively.
CL, AM and NB are medians of ABC.
C
The point where all 3 medians intersect
Centroid
Is the point of
concurrency
The centroid is 2/3’s of the distance
from the vertex to the side.
10
2x 32
5x
16
X
The centroid is the center of balance
for the triangle. You can
balance a triangle on the tip of
your pencil if you place the tip on
the centroid
angle bisector of a triangle
a segment that bisects an
angle of the triangle and goes
to the opposite side.
Any triangle has three angle bisectors.
B
E
A
In the figure, AF, DB and EC
F
MD
are angle bisectors of ABC.
C
Note: An angle bisector and a median of a triangle are
sometimes different.
Let M be the midpoint of AC. The median goes from
the vertex to the midpoint of the opposite side.
BM is a median
BD is a angle bisector of ABC.
The Incenter is where all
3 Angle bisectors intersect
Incenter
Is the point of concurency
Any point on an angle bisector is
equidistance from both sides of the angle
This makes the Incenter an
equidistance from all 3 sides
Let AD be a bisector of BAC,
M
B
P lie on AD,
PM  AB at M,
P
A
D
NP  AC at N.
N
C
Then P is equidistant from AB and AC.
Theorem: If a point lies on the bisector of
an angle, then the point is equidistant
from the sides of the angle.
Theorem: If a point lies on the bisector of
an angle, then the point is equidistant
from the sides of the angle.
The converse of this theorem is not always
true.
Theorem: If a point is in the interior of an
angle and is equidistant from the sides of
the angle, then the point lies on the
bisector of the angle.
Using the Angle Bisector Theorem
• What is the length of RM?
Because angle N has been bisected,
I know that each point along the
bisector is equidistant to the sides
Since MR and RP are both perpendicular
to each side and touch the bisector, I
know they are equal
7x = 2x + 25
5x = 25
x= 5
 What is the length of FB?
Because angle C has been bisected,
I know that each point along the
bisector is equidistant to the sides
Since BF and FD are both perpendicular
to each side and touch the bisector, I
know they are equal
6x +3 = 4x + 9
2x +3 = 9
2x = 6
x = 3
Definition of an Altitude of a Triangle
A altitude of a triangle is a segment that has one
endpoint at a vertex and the other creates a right angle at
the opposite side.
The altitude is perpendicular to the opposite side while
going through the vertex
Any triangle has three altitudes.
B
C
A
ACUTE
OBTUSE
Can a side of a triangle be its altitude? YES!
A
G
C
B
RIGHT
If ABC is a right triangle, identify its altitudes.
BG, AB and BC are its altitudes.
Orthocenter is where all the
altitudes intersect.
Orthocenter
The orthocenter can be located
in the triangle, on the triangle or
outside the triangle.
Obtuse
Right
Legs are altitudes
A Perpendicular bisector of a side does
not have to start at a vertex. It will form
a 90° angles and bisect the side.
Circumcenter
Is the point of concurrency
Any point on the perpendicular bisector
of a segment is equidistance from the
endpoints of the segment.
A
C
AB is the perpendicular
bisector of CD
D
B
This makes the Circumcenter an
equidistance from the 3 vertices
Perpendicular Bisector
Perpendicular Bisector
Using the Perpendicular Bisector
Theorem
• What is the length of AB?
Since BD perpendicular to the side
opposite B and bisects AC,
I know that BD is a perpendicular
bisector.
Since BD is a perpendicular
bisector, I know that BA and BC are
congruent since they are connected
to the vertex and the end of the
bisected line.
AB = 4x
AB = 4(5)
AB = 20
4x = 6x – 10
–2x = – 10
x=5
BC = 6x – 10
BC = 6(5) – 10
BC = 20
 What is the length of QR?
Since SQ is perpendicular to the
side opposite Q and bisects PR,
I know that SQ is a perpendicular
bisector.
Since SQ is a perpendicular
bisector, I know that PQ and QR are
congruent since they are connected
to the vertex and the end of the
bisected line.
PQ = 3n – 1
PQ = 3(3) –1
PQ = 8
3n – 1= 5n – 7
– 1= 2n – 7
6 = 2n
3=n
QR = 5(n) – 7
QR = 5(3) – 7
QR = 8
The Midsegment of a Triangle
is a segment that connects the midpoints of
two sides of the triangle.
The midsegment of a triangle is parallel to
the third side and is half as long as that side.
B
D
A
D and E are midpoints
E
DE is the midsegment
C
DE AC
1
DE  AC
2
Midsegment Theorem
The midsegment of a triangle is parallel to the third
side and is half as long as that side.
1
DE  AC
2
B
D
A
E
DE AC
C
1. Identify the 3 pairs of
parallel lines shown
above
UW
TX
WY
VT
YU
XV
2a.
If LK = 46,
what is NM ?
2b.
If JK = 5x + 20 and
NO = 20, find x
NO is half and big as JK
MN is half as long as LK
2(20) = 5x +20

40 = 5x + 20
2(MN) = 46
MN = 23
x=4
Example 1
In the diagram, ST and TU are midsegments of
triangle PQR. Find PR and TU.
16 ft
PR = ________
5 ft
TU = ________
Example 2
In the diagram, XZ and ZY are midsegments of
triangle LMN. Find MN and ZY.
53 cm
MN = ________
14 cm
ZY = ________
Example 3
In the diagram, ED and DF are midsegments of
triangle ABC. Find DF and AB.
2 (DF ) = AB
5X+2
2 (3x – 4 ) = 5x + 2
6x – 8 = 5x + 2
3X – 4
x–8= 2
x = 10
x = ________
10
DF = ________
26
AB = ________
52
Perpendicular Bisectors
• A point is equidistant from two objects if it is the same
distance from each.
Perpendicular Bisector Theorem: If a point is
on the perpendicular bisector of a segment,
then it is equidistant from the endpoints of
the segment.
Converse of the Perpendicular Bisector Theorem: If a point is
equidistant from the endpoints of a segment, then it is on the
perpendicular bisector of the segment.
Angle Bisectors
• The distance from a point to a line is the length of the
perpendicular segment from the point to the line.
Angle Bisector Theorem: If a point is on the
bisector of an angle, then the point is
equidistant from the sides of the angle.
Converse of the Angle Bisector 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.
There are 3 of each of these special
segments in a triangle.
The 3 segments are concurrent. They
intersect at the same point.
This point is called the point of
concurrency.
The points have special names and
special properties.
Altitude ..
Vertex .. 90° .. Orthocenter
Angle Bisector..
Angle into 2 equal angles .. Incenter
Perpendicular Bisector…
90° .. bisects side .. Circumcenter
Median ..
Vertex .. Midpoint of side ..Centroid
Give the best name for AB
A
|
A
|
B
B
Median Altitude
A
A
B
B
None Angle
Bisector
A
|
B
|
Perpendicular
Bisector
Survival Training
You’re Stranded On A Triangular
Shaped Island. The Rescue Ship Can
Only Dock On One Side Of The Island
But You Don’t Know Which Side. At
Which Point Of Concurrency Would
You Set Up Camp So You Are An Equal
Distance From All 3 Sides?
INCENTER
What If The Ship Could Only
Dock At One Of The Vertices?
Would You Change The
Location Of Your Camp ?
If So, Where?
YES CIRCUMCENTER
Where would you place a fire hydrant to
make it equidistance to the houses and
equidistance to the streets?
POST
Angle bisector for the streets
Perpendicular bisector for houses
POST