Download 5.1 Midsegment Theorem

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
1
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 
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
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
L
M
A
are a vertex and a
midpoint of the opposite
side.
B
N
C
The point where all 3 medians intersect
Centroid
Is the point of
concurrency
The centroid is 2/3 the distance
from the vertex to the side.
10
2x 32
5x
16
X
angle bisector of a triangle
a segment that bisects an
angle of the triangle and goes
to the opposite side.
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
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.
Acute Triangle
Orthocenter is where all the
altitudes intersect.
Orthocenter
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 equidistant from the
endpoints of the segment.
A
C
AB is the perpendicular
bisector of CD
D
B
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
Example 1
In the diagram, ST and TU are midsegments of
triangle PQR. Find PR and TU.
16 ft
PR = ________
5 ft
TU = ________
Give the best name for AB
A
|
A
|
B
B
Median Altitude
A
A
B
B
None Angle
Bisector
A
|
B
|
Perpendicular
Bisector