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Transcript
Theorems, Postulates, Definitions and Properties 5.1 Theorem - If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long. 5.2 Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segments. Theorem – If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Theorem – If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. Theorem – If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector. 5.3 Theorem – The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Theorem – The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle. A point is the circumcenter of a triangle if and only if it is the point of concurrency of the perpendicular bisectors of the triangle. A point is the incenter of a triangle if and only if it is the point of concurrency of the angle bisectors of the triangle. 5.4 A segment is a median of a triangle if and only if it is drawn from the vertex of a triangle to the midpoint of the opposite side. Theorem – The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. A point is the centroid of a triangle if and only if it is the point of concurrency of the medians of the triangle. A segment is an altitude of a triangle if and only if it is the perpendicular segment from the vertex of a triangle to the line containing the opposite side. Theorem – The lines that contain the altitudes of a triangle are concurrent. A point is the orthocenter of a triangle if and only if it is the point of concurrency of the altitudes of the triangle. 5.6 Property – If a = b and c > 0, then a > b. Theorem – The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles. Theorem – If two sides of a triangle are not congruent, then the larger angle lies opposite the larger side. Theorem – If two angles of triangle are not congruent, then the longer side lies opposite the larger angle. Theorem – The sum of the length of any two sides of a triangle is greater than the length of the third side. 5.7 Theorem – (SAS Inequality) - If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. Theorem – (SSS Inequality) - If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.