This chapter introduces different segments within triangles and how they relate to each other. We will explore the properties of midsegments, perpendicular bisectors, angle bisectors, medians, and altitudes. Next, we will look at the relationship of the sides of a triangle, how they relate to each other and how the sides of one triangle can compare to another.
This chapter begins with an introduction to the Midsegment Theorem. The definition of a perpendicular bisector is presented and the Perpendicular Bisector Theorem and its converse are explored. Now that the bisectors of segments have been discussed, the definition of an angle bisector is next and the Angle Bisector Theorem and its converse are presented. The properties of medians and altitudes of triangles are discussed in detail. The entire chapter builds to a discovery of the relationships between the angles and sides in triangles as a foundation for the Triangle Inequality Theorem. The chapter ends with a presentation of indirect proofs.
Perpendicular Bisector Theorem
Perpendicular Bisector Theorem Converse
Point of Concurrency
Concurrency of Perpendicular Bisectors Theorem
Angle Bisector Theorem
Angle Bisector Theorem Converse
Concurrency of Angle Bisectors Theorem
Concurrency of Medians Theorem
Triangle Inequality Theorem
SAS Inequality Theorem
SSS Inequality Theorem
If and are the midpoints of the sides they lie on, find:
The perpendicular bisector of .
The median of .
The angle bisector of .
Trace onto a piece of paper with the perpendicular bisector. Construct another perpendicular bisector. What is the point of concurrency called? Use this information to draw the appropriate circle.
Trace onto a piece of paper with the angle bisector. Construct another angle bisector. What is the point of concurrency called? Use this information to draw the appropriate circle.
Trace onto a piece of paper with the median. Construct another median. What is the point of concurrency called? What are its properties?
Trace onto a piece of paper with the altitude. Construct another altitude. What is the point of concurrency called? Which points of concurrency can lie outside a triangle?
A triangle has sides with length and . Find the range of the third side.
Texas Instruments Resources
In the CK-12 Texas Instruments Geometry FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9690.
This chapter continues to explore triangles with an emphasis on triangle relationships, properties of segments and angles, and the properties associated with the sides of triangles.