Download RGeo Ch 5 Study Guide

Regents Geometry
Chapter 5 Study Guide
Use the following website for on-line resources (practice tests and quizzes):
http://www.glencoe.com/sec/math/geometry/geo/geo_04/self_check_quiz/index.php/na
Select Chapter 5 Relationships in Triangles or use the resources found in the left-hand toolbar.
Chapter 5 – Relationships in Triangles
 Definitions with images can be found using Quizlet: http://quizlet.com/24603391/flashcards
Perpendicular Bisector of a Triangle
A line, segment, or ray that passes through the midpoint of the
side of a triangle and is perpendicular to that side.
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 a
segment, then it is on the perpendicular bisector of the segment.
Concurrent Lines
Point of Concurrency
Three or more lines that intersect at a common point.
The point where concurrent lines intersect.
Circumcenter
The point of concurrency of the perpendicular bisectors of a triangle, that is, the
point where the perpendicular bisectors intersect. This point can be on the
interior, exterior, or side of a triangle.
Circumcenter Theorem
The perpendicular bisectors of a triangle intersect at a point called the
circumcenter that is equidistant from the vertices of the triangle.
Bisector of an Angle
The locus of points in the interior of the angle equidistant from the sides
of the angle.
Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the
sides of the angle.
Converse of the Angle Bisector Theorem
If a point in the interior of an angle is equidistant from the
sides of the angle, then it is on the bisector of the angle.
Incenter
Incenter Theorem
Median of a Triangle
Centroid
The point of concurrency of the angle bisectors of a triangle, that is, the point
where the angle bisectors intersect.
The angle bisectors of a triangle intersect at a point called the incenter that is
equidistant from the sides of the triangle.
A segment whose endpoints are the vertex of a triangle and the midpoint of the
other side.
The point of concurrency of the medians of a triangle, that is, the point where
medians intersect.
Centroid Theorem
The medians of a triangle intersect at a point called the centroid that is two
thirds of the distance from each vertex to the midpoint of the opposite side.
Altitude of a Triangle
Orthocenter
A segment from a vertex to the the line containing the opposite side and
perpendicular to the line containing that side.
The point of concurrency of the altitudes of a triangle, that is, the point where the
altitudes intersect.
Direct Proof
A proof in which the statement (hypothesis) is assumed to be true. The laws of logic are
used to prove that the conclusion is true.
Indirect Proof A proof in which the statement (hypothesis) to be proved is assumed to be false and a
contradiction is shown. Also known as proof by contradiction.
Writing an Indirect Proof
1. Identify the conclusion you are asked to prove. Make the assumption
that this conclusion is false by assuming that the opposite is true.
2. Use logical reasoning to show that this assumption leads to a
contradiction of the hypothesis or some other fact.
3. Indicate that, since the assumption leads to a contradiction, the
original statement (what you were asked to prove) must be true.
Triangle Angle-Side Theorem
Triangle Inequality Theorem
1. If two sides of a triangle are not congruent, then the larger angle is
opposite the longer side.
2. If two angles of a triangle are not congruent, then the longer side is
opposite the larger angle.
The sum of the lengths of any two sides of a triangle must be greater
than the length of the third side.