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(5.1) Midsegments of Triangles
What will we be learning today?
Use properties of midsegments to
solve problems.
Theorem 5-1: Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the
segment is parallel to the third side, and is half its length
Key Terms:
A midsegment of a triangle is
connecting the midpoints of two sides.
a segment
Theorem 5-1: Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the
segment is parallel to the third side, and is half its length
Example 1: Finding Lengths
In
XYZ, M, N and P are the midpoints. The Perimeter of
and YZ.
MNP is 60. Find NP
Because the perimeter is 60, you can find NP.
x
NP + MN + MP = 60 (Definition of Perimeter)
NP +
NP +
+
= 60
24
M
= 60
P
22
NP =
Y
N
Z
Theorem 5-1: Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the
segment is parallel to the third side, and is half its length
Example 1:
Use the Triangle Midsegment Theorem to find YZ
MP =
of YZ Triangle Midsegment Thm.
MP = 24
24 = ½ YZ
= YZ
x
Substitute 24 for MP
Multiply both sides by 2
or the reciprocal of ½.
24
M
P
22
Y
N
Z
Theorem 5-1: Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the
segment is parallel to the third side, and is half its length
Example 2: Identifying Parallel Segments
Find the m<AMN and m<ANM. Line segments MN
and BC are cut by transversal AB, so m<AMN and <B
corresponding
are
angles.
A
Line Segments MN and BC are parallel by the
Triangle Midsegment
Theorem,
so m<AMN is congruent to <B by the
Corresponding Angles
Postulate.
m<AMN = 75 because congruent angles have the
same measure. In triangle AMN, AM = AN ,
so m<ANM = m<AMN by the Isosceles
Triangle Theorem. m<ANM = 75
by substitution.
N
M
M
75O
C
B
Theorem 5-1: Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the
segment is parallel to the third side, and is half its length
Quick Check:
1. AB = 10 and CD = 28. Find EB, BC, and AC.
A
E
D
B
C
Theorem 5-1: Triangle Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the
segment is parallel to the third side, and is half its length
Quick Check:
2. Critical Thinking Find the m<VUZ. Justify your answers.
X
65O
U
Y
V
Z
HOMEWORK
(5.4) Pgs. 325-363;
18- 26,27,49
(5.2) Bisectors in Triangles
What will we be learning today?
Use properties of perpendicular
bisectors and angle bisectors.
Theorems
Theorem 5-2:
Perpendicular Bisector
Thm.
Theorem 5-3: Converse of
the Perpendicular Bisector
Thm.
If a point is on the
perpendicular bisector of a
segment, then it is
equidistant form the
endpoints of the segment.
If a point is equidistant from
the endpoints of a segment,
then it is on the
perpendicular bisector of the
segment.
Theorems
Theorem 5-4:
Angle Bisector Thm.
Theorem 5-5:
Converse of the Angle
Bisector Thm.
If a point is on the bisector
of an angle, then it is
If a point in the interior of an
equidistant from the sides angle is equidistant from the
of the angle.
sides of the angle, then it is
on the angle bisector.
Key Concepts
The distance from a point to a line is the length of the
perpendicular segment from the point to the line.
Example:
D is 3 in. from line AB and line AC
C
D
3
A
B
Example
Using the Angle Bisector Thm.
Find x, FB and FD in the diagram at the right.
Show steps to find x, FB and FD:
FD =
7x – 35 = 2x + 5
A
2x + 5
B
F
C
Angle Bisector Thm.
D
7x - 35
E
Quick Check
a. According to the diagram, how far is K
from ray EH? From ray ED?
2xO
D
E
C
(X + 20)O
K
10
H
Quick Check
b. What can you conclude about ray EK?
2xO
D
E
C
(X + 20)O
K
10
H
Quick Check
c. Find the value of x.
2xO
D
E
C
(X + 20)O
K
10
H
Quick Check
d. Find m<DEH.
2xO
D
E
C
(X + 20)O
K
10
H
HOMEWORK
(5.2) Pgs. 267-269;
1-4, 6, 8-26, 28, 29,
40, 43, 46, 48