1. (5.1) Midsegments of Triangles
What will we be learning today?
Use properties of midsegments to solve problems.

2. Key Terms: A midsegment of a triangle is
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
a segment connecting the midpoints of two sides.

3. Example 1: Finding Lengths
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 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 M P 22 Y Z N

4. 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 = = ½ YZ Substitute 24 for MP = YZ Multiply both sides by or the reciprocal of ½. x 24 M P 22 Y Z N

5. Example 2: Identifying Parallel Segments
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 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. A
corresponding
Triangle Midsegment N M M
Corresponding Angles AN
m<AMN
Isosceles
75O C 75 B

6. AB = 10 and CD = 28. Find EB, BC, and AC.
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:
AB = 10 and CD = 28. Find EB, BC, and AC. A E B C D

7. 2. Critical Thinking Find the m<VUZ. Justify your answers.
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 Z Y V

8. HOMEWORK
(5.1) Pgs ;
1, 4, 6, 7-11, 13, 14, 18,
20-22, 26, 34, 36

9. (5.2) Bisectors in Triangles
What will we be learning today?
Use properties of perpendicular bisectors and angle bisectors.

10. Theorems
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.

11. 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.

12. 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

13. 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 = Angle Bisector Thm.
7x – 35 = 2x + 5 A
2x + 5 B F
7x - 35 C D E

14. 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)O10 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 m<DEH.
2xO D E C
(X + 20)O K 10 H

18. HOMEWORK
(5.2) Pgs ;
1-4, 6, 8-26, 28, 29,
40, 43, 46, 48