(5.1) midsegments of triangles what will we be learning today? use properties of midsegments to...
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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 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.
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 LengthsIn 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 =
24
22
x
P
Y
M
N Z
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 YZ
MP = of YZ Triangle Midsegment Thm.
MP = 24
24 = ½ YZ Substitute 24 for MP
= YZ Multiply both sides by 2 or the reciprocal of ½.
24
22
x
P
Y
M
N Z
Find the m<AMN and m<ANM. Line segments MN and BC are cut by transversal AB, so m<AMN and <B are angles.
Line Segments MN and BC are parallel by the Theorem, so m<AMN is congruent to <B by the
Postulate.
m<AMN = 75 because congruent angles have the same measure. In triangle AMN, AM = ,so m<ANM = by the Triangle Theorem. m<ANM = by substitution.
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
A
M
C
N
B75O
corresponding
Triangle Midsegment
Corresponding Angles
AN m<AMN Isosceles
75
M
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:
1. AB = 10 and CD = 28. Find EB, BC, and AC.A
C
B
D
E
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 m<VUZ. Justify your answers.
ZY
U
X
V
65O
(5.1) Pgs. 262-263;
1, 4, 6, 7-11, 13, 14, 18,
20-22, 26, 34, 36
HOMEWORK
(5.2) Bisectors in Triangles
What will we be learning today?
Use properties of perpendicular bisectors and angle bisectors.
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.
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.
Theorems
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.
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.
Theorems
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
Key Concepts
C
B
D
A3
Using the Angle Bisector Thm. Find x, FB and FD in the diagram at the right.
Example
A
E
F
CD
B 2x + 5
7x - 35
Show steps to find x, FB and FD:
FD = Angle Bisector Thm.
7x – 35 = 2x + 5
Quick Check
H
D
CE
(X + 20)O
a. According to the diagram, how far is K from ray EH? From ray ED?
2xO
K
10
Quick Check
H
D
CE
(X + 20)O
b. What can you conclude about ray EK?
2xO
K
10
Quick Check
H
D
CE
(X + 20)O
c. Find the value of x.
2xO
K
10
Quick Check
H
D
CE
(X + 20)O
d. Find m<DEH.
2xO
K
10
(5.2) Pgs. 267-269;
1-4, 6, 8-26, 28, 29,
40, 43, 46, 48
HOMEWORK