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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.