Journal Chapter 5

Perpendicular and Angle Bisectors

A perpendicular bisector is a a line that intersects at the midpoint of a segment forming four 90 degree angles.

Perpendicular Bisector theorem: If there is a point located on the perpendicular bisector of one segment, then it is equdistant from the endpoints of a segment.

A

B

CD

1) CA = DA2) DE = CE3) DF = CF

EF

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

X Y

A

B

E

XE = YEXD = YDXC = YC

Conclusion: AB is the perpendicular bisector of XY.

D

C

An angle bisector is a ray that intersects an angle creating two congruent angles.

Angle Bisector Theorem: If a point is located on the bisector of an angle, then it is equidistant from the sides of the angle.

L

MN

O P

Q

R S

T

U V

W

1) UV = VW2) OP = PQ3) RS = ST

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

A

B

CD

E

FG

H

I

AB = BCDE = EFGH = HI

Conclusion: W is the angle bisector.

W

Bisectors of Triangles

Concurrent: When more than two lines intersect at the same point.

The three perpendicular bisectors of a triangle meet at one point so that means that it is concurrent.

The point at which the prependicular bisectors of a triangle meet is called the circumcenter.

Incenter: The point of concurrency in which the angle bisectors of a traingle meet.

Since the three angle bisectors meet, they are concurrent.

Median: A segment that goes from one vertex of a triangle to the idpoint of the opposite segment.

When the three medians of the triangle intersect they form a point of concurrency called the centroid. This makes the medians concurrent.

Altitude: It is a line that is the perpendicular from one vertex to the opposite side of the triangle.

The point of concurrency in which the altitudes meet is called the orthocenter. Since the three lines intersect, they are concurrent.

orthocenter

orthocenter

Orthocenter

The Triangle Midsegment Theorem

Midsegment: A segment that connects the midpoints of two sides of a triangle.

Triangle Midsegment Theorem: A midsegment is parallel to a side of the triangle and it is also half of its lenght

A B

C D

AB ll CD

8

16 W

X

Y

Z

100

200

L M

N O

5000

10000

LM ll NO

Relationships Between Sides and Angles of a Triangle

In a triangle, the larger angle is opposite to the longer side. The smallest angle is opposite to the smallest side.

Biggest angle

Longest side

Smallest angle

Smallest side

8

20

5

The angles from smallest to largest:

A, C, B

A

B

C

10

8

4

Angles from largest to smellest:

M, N, L

L

M N

Triangle Inequalities

Triangle Inequalities Theorem: The sum of the lenghts of any two sides of a triangle is greater then the third side.

Exterior Angle Inequality: The exterior angle of a triangle is greater than the other angles in the triangle.

8 7

11

8 + 7 = 1511 + 7 = 1811 + 8 = 19

A

B C

A + C > BA + B > CB + C > A

A

B

C

DE

A + B > CA + C > B B + C > AD + E > CD + C > EC + E > D

Indirect ProofsSteps for creating an indirect proof:

1) Identify the given information.2) Assume the oposite of the given3) Prove what you assumed4) Prove that the assumption is false and find a contradiction.

Example: A scalene triangle cannot have two congruent sides.

A scalene triangle has two congruent sides. Assumption

Hinge Theorem

Hinge Theorem: When two sides in a triangle are congruent to two sides in another triangle and the included angles are not congruent, then the third side of the triangle that has a bigger included angle has a longer side.

Converse: If two sides of a triangle are congruent to two sides in another triangle, and the third sides are not congruent, the included angle that is across from the third side is bigger or smaller than the other triangle’s angle depending on the size of the third side.

A

B

C

D

E

F

BC > EF

1010

55

76

A

B

C

D

E

F

A > D

A

B

C

D

EF

F is bigger than C, so

ED > BA

Special Right Triangles45-45-90 Triangle Theorem: In a 45-45-90 triangle, both of the legs are congruent and the lenght of the hypotenuse is the lenght of a leg times the square root of 2.

4590

X8

X= 8 times the squre root of 2 90 45

15

X

X= 15 times the square root of 2

45

90

90 45

12

X

X= 12 times the square root of 2.

30-60-90 Triangle Theorem: In a 30-60-90 triangle, the lenght of the hypotenuse is two times the lenght of the smallest leg and the lenght of the longer leg is is the lenght of the smallest leg times the square root of 3.

90 30

60

A

B CY

12

X

AC = 24BC = 12 times the square root of 3

90

30

D

E

F

40

X

Y

X = 40 times the square root of 3Y = 80

90

60

L

M

NX

Y X = 200Y = 100 times the square root of 3.

100