Chapter 5, Section 1 Perpendiculars & Bisectors

Perpendicular Bisector A segment, ray, line or plane which is perpendicular to a segment at it’s midpoint is called a perpendicular bisector.

Equidistant Same length (distance).

Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of the segment, then it is on the perpendicular bisector of a segment.

Distance from a Point to a Line Is defined as the length of the perpendicular segment from the point to the line. When a point is the same distance from one line as it is to another line, then the point is equidistant from the two lines.

Angle Bisector Theorem If a point is on the angle bisector of an angle, then it is equidistant from the two sides of the angle.

Converse of the Angle Bisector Theorem If a point is in the interior of an angle and is equidistant from the two sides of the angle, then it lies on the bisector of an angle.

Chapter 5, Section 2 Bisector of a Triangle

Perpendicular Bisector of a Triangle Is a lie, ray or segment that is perpendicular to a side of the triangle at the midpoint of the side.

Concurrency Concurrent Lines: When three or more lines, rays or segments intersect at the same point. The point of intersection of the three lines is called the point of concurrency.

Concurrency of Perpendicular Bisectors of a Triangle The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle. The point of concurrency of the three perpendicular bisectors is called the circumcenter of the triangle.

Concurrency of Angle Bisectors of a Triangle The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. The point of concurrency of the angle bisectors is called the incenter of the triangle.

Chapter 5, Section 3 Medians and Altitudes of a Triangle

Median of a Triangle Is a segment whose endpoints are a vertex of the triangle & the midpoint of the opposite side. The point of concurrency of the medians is called the centroid of the triangle.

Concurrency of Medians of a Triangle

Altitudes of a Triangle Is the perpendicular segment from a vertex to the opposite side. Also, called the height. An altitude can lie inside, outside or on the triangle. The point of concurrency of the altitudes is called the orthocenter of the triangle.

Concurrency of Altitudes of a Triangle The lines containing the altitudes of a triangle are concurrent.

Chapter 5, Section 4 Midsegment Theorem

Midsegment of a Triangle A segment inside the triangle whose endpoints are at the midpoints of two sides of the triangle.

Midsegment Theorem

Inequalities in One Triangle Largest angle is across from the longest side Smallest angle is across from the shortest side

Theorems If one side is longer than another side of a triangle then the angle across from the longer side is larger than the angle across from the shorter side. If one angle is larger than another angle of a triangle then the side across from the larger angle is larger than the side across from the smallest angle.

Exterior Angle Inequality Theorem The measure of an exterior angle is larger than the measure of either of the two nonadjacent interior angles.

Triangle Inequality Theorem

Chapter 5, Section 5 Inequalities in two Triangles

Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

Converse of the Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.