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.

If is the bisector of then AC = BC.

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.

3 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

4 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.

6 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 bisects , that means

and , that means and since

bisects then

8 Geometry Lesson 5.1.notebook January 05, 2015

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