(5.1) Midsegments of Triangles

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(5.1) Midsegments of Triangles

• What will we be learning today?
• Use properties of midsegments to solve problems.

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Theorem 5-1 Triangle Midsegment TheoremIf 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
  

a segment
connecting the midpoints of two sides.

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Theorem 5-1 Triangle Midsegment TheoremIf 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 MNP is 60. Find NP and YZ.
• Because the perimeter is 60, you can find NP.
• NP MN MP 60 (Definition of Perimeter)
• NP 60 NP 60
• NP

x
24
P
M
22
Y
Z
N
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Theorem 5-1 Triangle Midsegment TheoremIf 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
  YZMP of YZ Triangle Midsegment Thm.MP
  2424 ½ YZ Substitute 24 for MP YZ
  Multiply both sides by 2
  or the reciprocal of ½.

x
24
P
M
22
Y
Z
N
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Theorem 5-1 Triangle Midsegment TheoremIf 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 mltAMN and mltANM. Line segments MN and BC
are cut by transversal AB, so mltAMN and ltB are
angles.Line Segments MN and BC are parallel
by the Theorem, so mltAMN is congruent to ltB
by the Postulate. mltAMN 75 because
congruent angles have the same measure. In
triangle AMN, AM ,so mltANM by the
Triangle Theorem. mltANM by substitution.

A
corresponding
Triangle Midsegment
M
N
M
Corresponding Angles
AN
mltAMN
Isosceles
75O
C
B
75
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Theorem 5-1 Triangle Midsegment TheoremIf 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
• AB 10 and CD 28. Find EB, BC, and AC.

A
E
B

C
D
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Theorem 5-1 Triangle Midsegment TheoremIf 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
mltVUZ. Justify your answers.

X
65O
U

Z
Y
V
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HOMEWORK

• (5.1) Pgs. 262-263
• 1, 4, 6, 7-11, 13, 14, 18,
• 20-22, 26, 34, 36

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(5.2) Bisectors in Triangles

• What will we be learning today?
• Use properties of perpendicular bisectors and
  angle bisectors.

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Theorem 5-2 Perpendicular Bisector Thm.If a
point is on the perpendicular bisector of a
segment, then it is equidistant form the
endpoints of the segment.
Theorems
Theorem 5-3 Converse of the Perpendicular
Bisector Thm.If a point is equidistant from the
endpoints of a segment, then it is on the
perpendicular bisector of the segment.

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Theorem 5-4 Angle Bisector Thm.If a point is
on the bisector of an angle, then it is
equidistant from the sides of the angle.
Theorems
Theorem 5-5 Converse of the Angle Bisector
Thm.If a point in the interior of an angle is
equidistant from the sides of the angle, then it
is on the angle bisector.

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The distance from a point to a line is the length
of the perpendicular segment from the point to
the line.
Key Concepts
Example D is 3 in. from line AB and line
AC
C
D
3
A
B
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Using the Angle Bisector Thm. Find x, FB and FD
in the diagram at the right.
Example
Show steps to find x, FB and FD FD
Angle Bisector Thm. 7x 35 2x 5

A
2x 5
B
F
7x - 35
C
D
E
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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
15
Quick Check
b. What can you conclude about ray EK?
2xO
D
E
C
(X 20)O
K
10
H
16
Quick Check
c. Find the value of x.
2xO
D
E
C
(X 20)O
K
10
H
17
Quick Check
d. Find mltDEH.
2xO
D
E
C
(X 20)O
K
10
H
18
HOMEWORK

• (5.2) Pgs. 267-269
• 1-4, 6, 8-26, 28, 29,
• 40, 43, 46, 48