1. Angle Bisectors and Perpendicular Bisectors
Section 5.6

2. Objectives
Use angle bisectors and perpendicular bisectors.

3. Key Vocabulary Distance from a point to a line. Equidistant
Angle bisector (review)
Perpendicular bisector

4. Theorems
5.3 Angle Bisector Theorem
5.4 Perpendicular Bisector Theorem

5. Angle Bisector
A bisector which we have already studied is an  bisector.
Definition: Angle Bisector - A ray that divides an angle into two congruent angles.
In the picture to the right, the red line segment is the angle bisector. The  arc marks show the 2  angles that were formed when the angle bisector bisected the original angle.

6. Angle Bisectors An  bisector divides an  into two ≅ parts.
In a ∆, an  bisector divides one of the ∆’s s into two ≅ s. (i.e. if AD is an  bisector then BAD ≅ CAD)
Angle Bisector

7. Angle Bisectors Review – Distance means ⊥ distance.
Distance from a point to a line is the length of the  segment from the point to the line.
PD=distance from P to line l. P l D

8. Angle Bisectors
When a point is the same distance from two or more objects, the point is said to be equidistant from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points.
A locus is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment.

9. Angle Bisectors
The bisector of an angle can be described as the locus of points in the interior of the angle equidistant from the sides of the angle.
This description leads to the following theorems.

10. Angle Bisectors Theorem 5.3 – Angle Bisector Theorem) O D G A
If a pt is on the bisector of an , then it is equidistant from the sides of the .

11. UW bisects TUV. ∆UTW and ∆UVW are right triangles.
Example 1
Use the Angle Bisector Theorem
Prove that ∆TWU  ∆VWU.
∆TWU  ∆VWU.
UW bisects TUV. ∆UTW and ∆UVW are right triangles.
SOLUTION
Statements
Reasons 1.
Given
UW bisects TUV.
Given 2.
∆UTW and ∆UVW are right triangles.
Reflexive Prop. of Congruence 3.
WU  WU
Angle Bisector Theorem 4.
WV  WT
HL Congruence Theorem 5.
∆TWU  ∆VWU

12. Example 2: Applying the Angle Bisector Theorem
Find the measure. BC
BC = DC
 Bisector Thm.
BC = 7.2
Substitute 7.2 for DC.

13. Your Turn: Given that YW bisects XYZ and WZ = 3.05, find WX. WX = WZ
 Bisector Thm.
WX = 3.05
Substitute 3.05 for WZ.
So WX = 3.05

14. Your Turn:
Find the measure of SR.
A. 22
B. 5.5
C. 11
D. 2.25

15. Your Turn:
C. Find the measure of UV.
A. 7
B. 14
C. 19
D. 25

16. Example 3: Using Angle Bisectors
Roof Trusses: Some roofs are built with wooden trusses that are assembled in a factory and shipped to the building site. In the diagram of the roof trusses shown, you are given that AB bisects CAD and that ACB and ADB are right angles. What can you say about BC and BD?

17. SOLUTION:
Because BC and BD meet AC and AD at right angles, they are perpendicular segments to the sides of CAD. This implies that their lengths represent distances from the point B to AC and AD. Because point B is on the bisector of CAD, it is equidistant from the sides of the angle.
So, BC = BD, and you can conclude that BC ≅ BD.

18. Review Definitions
Midpoint: The midpoint of a segment is the point halfway between the endpoints of the segment. If X is the midpoint of AB, then AX = XB.
Example:
Segment Bisector: Any segment, line, or plane that intersects a segment at its midpoint is called a segment bisector. X A B V S E T O

19. Perpendicular Bisector
Given segment
perpendicular bisector P A B C
midpoint
A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector.
CP is a  bisector of AB

20. Perpendicular Bisector
Theorem 5.4 Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. A B C D

21. Use the diagram to find AB.
Example 4
Use Perpendicular Bisectors
Use the diagram to find AB.
SOLUTION
In the diagram, AC is the perpendicular bisector of DB.
8x = 5x +12
By the Perpendicular Bisector Theorem, AB = AD.
3x = 12
Subtract 5x from each side. 2 3x 3 12 =
Divide each side by 3.
x = 4
Simplify.
You are asked to find AB, not just the value of x.
ANSWER
AB = 8x = 8 · 4 = 32

22. Triangle Perpendicular Bisector
Side AB
perpendicular bisector
of ∆ABC P A B C
A ⊥ bisector of a ∆ is
a line, segment, or
ray that passes
through the midpoint
of one of the sides of
the ∆ at a right angle.

23. Triangle Perpendicular Bisector
The perpendicular bisector of a side of a triangle does not necessarily pass through a vertex of the triangle.
Example: In ∆ABC, the red line segment is the ⊥ bisector side AC and does not pass through vertex B.

24. Triangle Perpendicular Bisector
Properties:
Any point on the perpendicular bisector of a segment is equidistant
from the endpoints of the segment.
Any point equidistant from the endpoints of a segment lies on the
perpendicular bisector of the segment.
Both sides are congruent- make sure you see this or it is NOT a perpendicular bisector
Perpendicular Bisector
Two conditions: midpoint and perpendicular

25. Triangle Perpendicular Bisector
Tell whether each red segment is an perpendicular bisector of the triangle. NO NO
YES

26. Example 5: Applying the Perpendicular Bisector Theorem
Find each measure. MN
MN = LN
 Bisector Thm.
MN = 2.6
Substitute 2.6 for LN.

27. Example 6: Applying the Perpendicular Bisector Theorem
Find each measure. TU
TU = UV
 Bisector Thm
3x + 9 = 7x – 17
Substitute the given values
9 = 4x – 17
Subtract 3x from both sides
26 = 4x
Add 17 to both sides
6.5 = x
Divide both sides by 4
So TU = 3(6.5) + 9 = 28.5

28. Your Turn: Find the measure.
Given that line ℓ is the perpendicular bisector of DE and EG = 14.6, find DG.
DG = EG
 Bisector Thm.
DG = 14.6
Substitute 14.6 for EG.

29. Your Turn:
A. Find the measure of NO.
A. 4.6
B. 9.2
C. 18.4
D. 36.8

30. Concurrent Lines
When two lines intersect at one point, we say that the lines
are intersecting. The point at which they intersect is
the point of intersection.
(nothing new right?)
Well, if three or more lines intersect at a common point, we say that the lines are concurrent lines. The point at which these lines intersect is called the point of concurrency.

31. Definitions
Concurrent Lines – Three or more lines that intersect at a common point.
Point of Concurrency – The point where concurrent lines intersect.

32. Perpendicular Bisector and Concurrent Lines
A triangle has 3 sides, so it
also has 3 perpendicular
bisectors, one for each side.
The 3 perpendicular
bisectors are concurrent
lines.
The point of concurrency of
the perpendicular bisectors
is called the circumcenter of
the triangle.
In the picture to the below
point K is the circumcenter.

33. Circumcenter The point of concurrency of the three perpendicular
bisectors of a triangle is the
circumcenter.
The circumcenter of a
triangle is one of many
different “centers” of a
triangle.
Circumcenter

34. Circumcenter
The circumcenter is equidistant from all three Vertices.

35. Circumcenter The circumcenter gets its name from the fact that
it is the center of
the circle that
circumscribes
the triangle.
Circumscribe
means to be
drawn around a
figure and
passing through
each vertex.

36. Circumcenter
To find the center of a circle that will circumscribe any given triangle, find the point of concurrency of the three perpendicular bisectors of the triangle, the circumcenter. Sometimes this will be inside the triangle, sometimes it will be on the triangle, and sometimes it will be outside of the triangle!
Circumcenter
Acute ∆ - inside
Obtuse ∆ - outside
Circumcenter
Right ∆ - on

37. Circumcenter
The Perpendicular Bisectors of a triangle intersect at a point that is equidistant from the vertices of a triangle.
Example: If P is the circumcenter of ∆ABC, then
PA = PB = PC.

38. will be equidistant from each store.
Example 7
Use Intersecting Bisectors of a Triangle
A company plans to build a warehouse that is equidistant from each of its three stores, A, B, and C. Where should the warehouse be built?
SOLUTION
Think of the stores as the vertices of a triangle. The point where the perpendicular bisectors intersect
will be equidistant from each store.
Trace the location of the stores on a piece of paper. Connect the points of the locations to form ∆ABC. 1. 38

39. Draw the perpendicular bisectors
Example 7
Use Intersecting Bisectors of a Triangle 2.
Draw the perpendicular bisectors
of AB, BC, and CA. Label the intersection of the bisectors P.
ANSWER
Because P is equidistant from each vertex of ∆ABC, the warehouse should be built near location P. 39

40. Your Turn:
3 principals have a meeting. They want to meet at a central location so everyone travels the exact same distance.
How would you find the central location?
2. Why?
Find the circumcenter
2. Same distance from the 3 vertices.

41. Practice

42. 1. Find FH. ANSWER 5 2. Find MK. ANSWER 20 3. Find EF. ANSWER 15
Use Angle Bisectors and Perpendicular Bisectors 1.
Find FH.
ANSWER 5 2.
Find MK.
ANSWER 20 3.
Find EF.
ANSWER 15

43. Practice
No, right ∠ not indicated.

44. Practice
No, right ∠ not indicated.

45. Practice
Yes, right ∠ and midpoint
are indicated.

46. Practice
No, equidistant of
Point P not indicated.

47. Practice No, segments from sides to point P not indicated
as perpendicular.

48. Practice
m∠1=55
m∠2=55
m∠3=20
m∠4=70
m∠5=35
m∠6=50

49. Practice
m∠1=25
m∠2=65
m∠3=65
m∠4=115
m∠5=115
m∠6=20
m∠7=45
m∠8=45

50. Joke Time How can you get four suits for a dollar?
Buy a deck of cards.
Guy buys a parrot that is constantly using foul language. Really horrible stuff. Finally the guy gets fed up and throws the parrot in the freezer to punish him. After about an hour, he hears a faint tapping sound from inside the freezer and opens the door. There’s the parrot, wings wrapped around himself, shivering. He says,
“I swear, I’ll never, ever curse again. But can I ask you a question? What did the chicken do?"

51. Assignment
Pg. 276 – 280: #1 – 25 odd, 33 – 37 odd, 33, 35, 45, 47