1. 5.1 and 5.2: Midsegments of Triangles Perpendicular and Angle bisectors
Objectives: Students will be able to…
Use properties of midsegments to solve problems
Use properties of perpendicular and angle bisectors to find missing measurements

2. Midsegment of a Triangle
Segment connecting the midpoints of 2 sides of a triangle B D E C A
D is the midpoint of
E is the midpoint of
is the midsegment of

3. Triangle Midsegment Theorem
If a segment joins the midpoints of 2 sides of a triangle, the segment is parallel to the 3rd side, and is ½ its length
Do NOT assume it’s a midsegment unless they tell you or you prove it.

4. Triangle Midsegment Theorem
is the midsegment of
Therefore….
AND

5. EXAMPLES: Find the value of the variables.1. 2. A B C x
x+2 E D 18 20

6. Find the perimeter of D 5 3 E 7 A

7. Find the value of the variable.
(6x)°
30°

8. In ∆XYZ, M, N, and P are midpoints. The perimeter of the ∆ MNP is 60 yd. Find NP and YZ.22 M P 24
NAME ALL PARALLEL
SEGMENTS: Y Z N

9. What is the measure of angle ANM? Angle A? Explain.
65° C B

10. Warm Up What is a perpendicular bisector of a segment?
What is an angle bisector?
What does equidistant mean?

11. is the perpendicular bisector of
What do we know?

12. Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment

13. Converse of Perpendicular Bisector Theorem
If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
IS THE PERPENDICULAR BISECTOR OF SEGMENT AB 6 6

14. EXAMPLES
Find PB and AQ. 14 7

15. Find AD, x, and BC. 12 C D A
2x+6
3x+1 B

16. What do we know about P? 10 10

17. FYI
When we refer to the distance from a point to a line, we are talking about the length of the perpendicular segment from point to a line.

18. Angle Bisector Theorem
If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. (perpendicular distance to sides is the same) 4 4

19. Converse of Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

20. You are designing a park, and you are in charge of building a walkway where every point on the walkway will be equidistant from 2 major monuments in the park. How would you figure out where to put the walkway?