Chapter 5 Relationships in Triangles

__Section 5.1 Bisectors, Medians, and
Altitudes__

**Perpendicular Bisector: **A line, segment or ray
that intersects the midpoint of a segment at a right angle.

**Theorem 5.1:** Any point on the perpendicular
bisector of a segment is equidistant from the endpoints of the segment.

**Theorem 5.2:** Any point equidistant from the
endpoints of the segment lies on the perpendicular bisector of the
segment. (this is the converse of Theorem 5.1)

**Concurrent Lines:** Three or more lines that
intersect at a common point.

**Point of Concurrency: ** The point of intersection
of concurrent lines.

**Angle Bisector: **A ray that divides an angle
into two congruent angles.

**Theorem 5.4:** Any point on the angle bisector is
equidistant from the sides of the angle.

**Theorem 5.5:** Any point equidistant from the sides
of an angle lies on the angle bisector. (this is the
converse of Theorem 5.4)

__ Perpendicular Bisectors of a Triangle__:
run through the
midpoint of a side, perpendicular to that side

Point of Concurrency of the lines
is the **circumcenter**, which is ** equidistant to the vertices** of the triangle. (this
can be shown with a circumcircle)

** Angle Bisectors: ** bisects the angles of the
triangle.

Point of Concurrency of the lines
is the **incenter**, which is ** equidistant to the sides** of the triangle. (this can be
shown with an incircle)

__ Medians__: runs from the midpoint of a side to
the opposite vertex.

Point of Concurrency of the lines
is the **centroid**, which is the center of the surface of the triangle.

The distance from centroid to vertex is twice the distance from centroid to midpoint.

__ Altitudes__: segments from the vertex of the
triangle in a direction perpendicular to the opposite side

Point of Concurrency of the segments is the orthocenter.

** Mid-Segments**: Segments
that connect the midpoint of one side of a triangle with a midpoint of another
side

Each mid-Segment is parallel to the side of the original triangle that it does not intersect. (For example FD is parallel to CB)

Each mid-Segment is 1/2 the length of the side of the original triangle that it does not intersect. (For example if CB=20, then FD =___)

Find the length of each segment

__Section 5.2 Inequalities and Triangles__

__Triangle
Inequalities__

The side of a triangle opposite the largest angle is the longest side.

The side of a triangle opposite the smallest angle is the shortest side.

The side of a triangle opposite the mid-sized angle is the medium length side.

List the sides in order from longest to shortest.

The angle of a triangle opposite the longest side is the largest angle.

The angle of a triangle opposite the shortest side is the smallest angle.

The angle of a triangle opposite the mid-sized side is the mid-sized angle.

List the angles in order from least to greatest.

List the sides in order from least to greatest

__ Exterior Angle
Inequality Theorem:__ An exterior angle of a
triangle is greater than either of its remote interior angles. (

__Section 5.4 The Triangle Inequality__

**Theorem 5.11:** The sum of the lengths of any two sides of a triangle is
greater than the length of the third side.

**To simplify the concept of Th 5.11, the
sum of the two shortest sides must exceed the longest side.**

**Example 1: Can a triangle have
sides of 4, 10,and 5?
**

**Example 2: Can a triangle have
sides of 3, 7, and 5?
**

**Find the range of possibilities for the third side of a
triangle:**

**5 and 8**

**10 and 4**

**6 and 6**