Geometry Lesson 5.1.notebook
January 05, 2015
Which congruence, if any, determines whether the following triangles are congruent?
a.
b.
c.
1
Geometry Lesson 5.1.notebook
January 05, 2015
Bisectors of Triangles
Perpendicular Bisector Theorem ­ If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.
is the bisector of
If
AC = BC.
then
Converse of the Perpendicular Bisector Theorem ­ If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.
If AC=BC, then C lies on
, the perpendicular bisector of 2
Geometry Lesson 5.1.notebook
January 05, 2015
Find each measure.
Concurrent Lines ­ Three or more lines intersect at a common point.
Point of Concurrency ­ The point where concurrent lines intersect.
a, b, and c are concurrent at P.
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Geometry Lesson 5.1.notebook
January 05, 2015
The three perpendicular bisectors of a triangle are concurrent lines. The point of concurrency of the perpendicular bisectors is called the circumcenter. The circumcenter can be on the interior, exterior, or side of a triangle.
Acute triangle
Obtuse triangle
Right Triangle
The Circumcenter Theorem ­ The perpendicular bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle.
If P is the circumcenter of then PA = PB = PC.
Real World Example 2: P. 324
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Geometry Lesson 5.1.notebook
January 05, 2015
Angle Bisectors
Angle Bisector Theorem ­ If a point is on the bisector of an angle, then it is equidistantfrom the sides of an angle.
If
bisects then DF = FE.
Converse of the Angle Bisector Theorem ­ If the point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.
If then bisects 5
Geometry Lesson 5.1.notebook
January 05, 2015
Find each measure.
A triangle has three angle bisectors. Theses bisectors are also concurrent. The point of concurrency is called the incenter.
P is the point of concurrency or the incenter of triangle ABC.
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Geometry Lesson 5.1.notebook
January 05, 2015
Incenter Theorem ­ The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from each side of the triangle.
If P is the incenter of then 7
Geometry Lesson 5.1.notebook
January 05, 2015
Find each measure if J is the incenter of triangle ABC.
a. JF Since JF=JE=JD, we can use the Pythagorean Theorem
to find JE.
Since JE = 9, then JF = 9.
b. Because bisects
and
, that means bisects
and bisects
, that means and since
then 8
Geometry Lesson 5.1.notebook
January 05, 2015
P. 327 9­35
9