(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