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 P. 327 9­35 9.