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GEOMETRY JOURNAL 5. MELANIE DOUGHERTY. Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse. A perpendicular bisector is a line perpendicular to the base of a triangle that bisects it. Perpendicular Bisector theorem:

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Geometry journal 5

GEOMETRY JOURNAL 5

MELANIE DOUGHERTY


Describe what a perpendicular bisector is. Explain the perpendicular bisector theorem and its converse.

  • A perpendicular bisector is a line perpendicular to the base of a triangle that bisects it.

  • Perpendicular Bisector theorem:

    • If a point is on the perpendicular bisector of a segment, then it is equidistant form the endpoints of the segment.

  • Converse:

    • if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.


Perpendicular bisector examples
Perpendicular Bisector Examples perpendicular bisector theorem and its converse

AB = AC


AB = AC perpendicular bisector theorem and its converse


AC = BC perpendicular bisector theorem and its converse


Pb converse examples
PB Converse Examples perpendicular bisector theorem and its converse

LN = EN


AD = DC perpendicular bisector theorem and its converse


CD = DB perpendicular bisector theorem and its converse


Describe what an angle bisector is explain the angle bisector theorem and its converse
Describe what an angle bisector is. Explain the angle bisector theorem and its converse. 

  • An angle bisector is a line that divides the angle.

  • The angle Bisector theorem:

    • If a point is on the bisector of an angle, then it is equidistant from the sides of the angle

  • Converse:

    • If a point is equidistant from the sides of an angle the it is on the bisector.


Angle bisector theorem examples
Angle Bisector theorem examples bisector theorem and its converse. 

BF = FC


<UFK is congruent to <KFC bisector theorem and its converse. 


<EWR is congruent to <RWT bisector theorem and its converse. 


Concurrent
CONCURRENT bisector theorem and its converse. 

When 3 or more lines intersect at one point


Concurrency of perpendicular bisector theorem of triangles
Concurrency of perpendicular bisector theorem of triangles bisector theorem and its converse. 

The circumcenter of a triangleisequidistant from the vertices of the triangle.

Circumcenter: where the 3 perpendicular bisectors of a triangle meet

circumcenter

circumcenter

circumcenter


acute bisector theorem and its converse. 

DA = DB = DC

right

DA = DB = DC

DA = DB = DC

obtuse


Concurrency of angle bisectors of a triangle theorem
concurrency of angle bisectors of a triangle theorem bisector theorem and its converse. 

Incenter of a triangle : where the 3 angle bisectors of a triangle meet

Concurrency of a angle bisectors of a triangle theorem: the incenter of a triangle is equidistant from the sides of the triangle.

incenter

incenter

incenter


ACUTE bisector theorem and its converse. 

RIGHT

DF = DG = DE

DF = DG = DE

DF = DG = DE

OBTUSE


Medians and altitudes of triangles
MEDIANS AND ALTITUDES OF TRIANGLES bisector theorem and its converse. 

The median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side

The centroid of a triangle is the point of concurrency of the medians of a triangle.

Concurrency of medians of a triangle theorem: the centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side.


Examples
EXAMPLES bisector theorem and its converse. 

CMTT

CENTROID

MEDIAN


Concurrency of altitudes of triangles theorem
Concurrency of altitudes of triangles theorem bisector theorem and its converse. 

Altitude: a perpendicular segment from a vertex to the line containing the opposite side

Orthocenter: point where the 3 altitudes of a triangle meet.

Concurrency of altitudes of triangles theorem: the lines containing the altitude are concurrent


Triangle midsegment theorem
Triangle Midsegment theorem bisector theorem and its converse. 

A midsegment is a segment that joins the midpoints of two sides of a triangle

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

midsegment

midsegment

midsegment


AB bisector theorem and its converse. ll EF, EF = ½ AB

DE ll BC, DE = ½ BC

DE ll AC, DE = ½ AC


Angle side relationship in triangles
Angle-Side Relationship in Triangles bisector theorem and its converse. 

If none of the sides of the triangle are congruent then the largest side is opposite the largest angle.

If none of the sides of the triangle are congruent then the shortest side is opposite the smallest angle.


Examples1
EXAMPLES bisector theorem and its converse. 


Triangle inequality
Triangle Inequality bisector theorem and its converse. 

The sum of the lengths of two sides of a triangle is greater than the length of the third side.


Writing an indirect proof
Writing an indirect proof bisector theorem and its converse. 

Identify what is being proven

Assume that the opposite of your conclusion is true

Use direct reasoning to prove that the assumption has a contradiction

Assume that if the 1st assumption is false then what is being proved is true.


Examples2
EXAMPLES bisector theorem and its converse. 

Step 1

Given: triangle JKL is a right triangle

Prove: triangle JKL doesn't have and obtuse angle

Step 2

Assume <K is an obtuse angle

Step 3

m<K + m<L = 90

m<K = 90 – m<L

m<K > 90

90 – m<L > 90

m<L <0 (this is impossible)

Step 4

The original conjecture is true.


Hinge theorem
Hinge theorem bisector theorem and its converse. 

If 2 sides of a triangle are congruent to 2 sides of an other triangle and included angles are not congruent, then the longer third side is across from the larger included angle.

Converse: if 2 sides of 2 triangle are congruent to 2 sides of an other triangle and the third sides of an other triangle are not congruent, then the larger included angle is across from the longer third side.


Examples3
Examples bisector theorem and its converse. 


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