Chapter 5 Journal

By: Ana Cristina Andrade

Perpendicular Bisector:Definition of Perpendicular bisector: a line perpendicular to a segment at the line segment´s midpoint

Perpendicular bisector theorem: If a line is perpendicular, then it is equidistant from the endpoints of a segment.

Converse of perpendicular bisector theorem: If a point is equidistant from the endpoint of a segment, then it is perpendicular line.

Examples:

1. =2. =3. =

Perpendicular bisector theorem

Examples:Converse of perpendicular

bisector theorem

Angle Bisector:Angle bisector theorem: a ray or line that cuts an angle into 2 congruent angles. It always lies on the inside of an angle

Converse of angle bisector theorem: If a point is equidistant from the sides of a angle, then it lies on the bisector angle.

Examples: Angle bisector theorem

AB

C

A

B

CAB

C

AB = CB

Examples:Converse of

angle bisector theorem

A

B

C

D

<ADB = <CDB(Congruent ,not equal)

A

B

C

D

A

B

C

D

Concurrency:Definition of concurrency: Where three or more lines intersect at one point.

concurrency of Perpendicular bisectors:Concurrency of perpendicular bisectors: Point where the perpendicular bisectors intersect.

Circumcenter:Definition of Circumcenter: the point of congruency where the perpendicular bisectors of a triangle meet.The circumcenter theorem: The circumcenter of a triangle is equidistant from the vertices of the triangle.

Examples:

concurrency of angle bisectors:Concurrency of angle bisectors: Point where the angle bisectors intersect.

Incenter:Definition of incenter: The point where the angle bisector intersect of a triangleAlways occur on the side of triangleIncenter theorem: The incenter of a triangle is equidistant from the side of a triangle

Examples:

Median:Definition of Median: segment that goes from the vertex of a triangle to the opposite midpoint.

Centroid:Centroid: The point where the medians of a triangle intersect.The distance from the vertex to the centroid is double the distance from the centroid to the opposite midpoint.

Examples:

concurrency of medians:Concurrency of medians: point where the medians intersect.

Altitude:

Definition of altitude: a segment that goes from the vertex perpendicular to the line containing the opposite side.

Examples:

Orthocenter:Definition of Orthocenter: Where the altitudes intersectIf the triangle is acute, the orthocenter is on the inside of the triangleIf it is right orthocenter is on the vertex of the right angle.

Examples:

concurrency of altitudes:Concurrency of altitudes: point where the altitudes intersect.

Midsegment:Midsegment of a triangle: segment that joins the midpoints of two sides of the triangleA midsegment of a triangle, and its length is half the length of that side.

Examples:

midsegment theorem:Triangle midsegment theorem: A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.

relationship between the longer and shorter sides of a triangle:

Hinge theorem: If 2 triangles have 2 sides that are congruent, but the third side is not congruent, then the triangle with the larger included angle has the longer third side.Converse of Hinge theorem: If two sides of a triangle are congruent to the two sides of the other triangle but the other sides are not congruent then, the largest included angle is across from the largest side.

Examples:

<B > <Y

AC > XZ

A

B

C

H

I

J

AB

C

H

I J

J>A

HI>B

C

Hinge Theorem

Examples: Converse of Hinge Theorem

A

B

C

D

E

F

FE > CB, FD=CA, DE = AB (congruent)

<D> <A

A

BC

D

F E

B

ED

AC

FE

Relationship between opposite angles of a triangle:

Triangle side-angle relationship theorem: In any triangle, the longest side is always opposite from the largest angle and vice versa.

Examples:

Longest side

Shortest side

Longest side

exterior angle inequality:The non-adjacent interior angles are smaller than the exterior angleA+B = exterior angle (c)

A B CA

CB

A

BC

Triangle inequality:

Triangle inequality theorem: the 2 smaller sides of a triangle must add up to more than the length of the 3rd side.

Examples:4, 7, 104+7=11

YES2, 9, 122+9=11

NO

3, 1.1, 1.71.1+1.7= 2.8

NO

indirect proof:Indirect proof: used when it is not possible to prove something directly.

Steps:1.Assume that what you are proving is false2.Use that as your given, and start proving it3.When you come to a contradiction you have

proved that it is true.

Examples:Prove: A triangle cannot have 2 right angles

A triangle has 2 right angles (<1 & <2)

Given

M<1=m<2=90 Def. right angle

M<1+m<2=180 Substitution

M<1+m<2+m<3=180 Triangle sum theorem

M<3=0 contradiction

Examples:Proove: a right triangle cannot have an obtuse angle

A right triangle can have an obtuse angle (<A)

Given

M<A + m<B= 90 Substitution

M<A =90 – m<b Subtraction prop.

M<A> 90° Def. obtuse triangle

90° - m<b > 90 substitution

m<b = 0 contradiction

Examples:A triangle cannot have 4 sides

A triangle can have 4 sides

Given

A square is a shape with 4 sides

Def of square

A triangle is a shape with only 3 sides

Def of triangle

A triangle cannot have 4 sides

contradiction

special relationships in the special right triangles:

45° - 45° - 90° triangle theorem: In this kind of triangle, both legs are congruent and the hypotenuse is the length of a leg times √230° - 60° - 90° Triangle theorem: In this kind of triangle the longest leg is √3 the shorter leg and the hypotenuse is √2 the shortest side of the triangle.

Examples:

X

X

BC=AC=XAB=X√2

A

B C

45° - 45° - 90° triangle theorem

Examples:

45°

45°

14X

X=14√2

45° - 45° - 90° triangle theorem

Examples:30° - 60° - 90° triangle theorem

B16

16=2a8=aB=a√3B=8√3

Y

2020=2x10=xY=a√3Y=10√3

d

100

100=2d50=dH=d√3H=50√3