Perpendiculars and BisectorsOBJECTIVE: To use properties of perpendicular bisectors and angle bisectors

BIG IDEAS:Reasoning and Proof Measurement

ESSENTIAL UNDERSTANDINGS: • There is a special relationship between the points on the perpendicular

bisector of a segment and the endpoints of the segment.• There is a special relationship between the points on the bisector of an angle

and the sides of the angle.

MATHEMATICAL PRACTICE:Look for and express regularity in repeated reasoning

GETTING READY• You hang a bulletin board over your desk using string. The

bulletin board is crooked. When you straighten the bulletin board, what type of triangle does the string form with the top of the board? How do you know?

• Visualize the vertical line along the wall that passes through the nail. What relationships exist between this line and the top edge of the straightened bulletin board? Explain

Terms • Perpendicular bisector: a ____________________, ray,

_______________, or plane that is ____________________ to a segment at its ____________________

• Equidistant from two points: the _______________ distance from

_______________ point as from another point.

• Perpendicular Bisector Theorem: If a point is on the ____________________ _______________ of a segment, then it is ____________________ from the _______________ of the segment.

• Converse of the Perpendicular Bisector Theorem: If a point is

____________________ from the _______________ of a segment, then it is on the ____________________ _______________ of the segment.

Example

• EX 1: In the diagram shown, is the perpendicular bisector of

• a) What segment lengths in the diagram are equal?

• b) Explain why T is on

PQ�

CD

PQ�

T

7 7

C Q D

P

Example

• EX 2: In the diagram, is the perpendicular bisector of

• a) Find the value of x

• b) Find the value of y

• c) Is E on ? Explain

AB�

CD

AB�

C 14 9 xE A B

15 6 y

D

Terms • Distance from a point to a line: the _______________ of the

____________________ segment from the point to the _______________

• Equidistant from two lines: the _______________ distance from one _______________ as from another line.

• Angle Bisector Theorem: If a _______________ is on the ____________________ of an _______________, then it is ____________________ from the two __________ of the angle.

• Converse of the Angle Bisector Theorem: If a _______________ is in the interior of an _______________ and is ____________________ from the sides of the angle, then it lies on the _______________ of the angle.

Example• EX 3: Determine the correct measurement for the angle or

segment given. A• a)

D E F • b) B

C G

• c)

DCB

FE

AC

55 8

20

Example

• EX 4: Determine the correct measurements for the angle or segment given. B

A • a) E 6 C

10 • b) 8 G

F 3 D• c)

H• d)

• e)

EG

GDE

ED

HD

FD

40

73

Proof

• GIVEN:• PROVE:

C

A D B

is on the perpendicular bisector of C ABADC BDC