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Chapter 5 Test (5.1-5.5 Skip 5.4) Section 1: Midsegments of a Triangle Midsegment- a segment connecting the midpoints of two sides of a triangle. A midsegment is both parallel and half of the third side Section 2: Bisectors in Triangles Perpendicular Bisectors o A line, segment, or ray that is perpendicular to the segment at its midpoint. o Perpendicular Bisector Theorem-If a point is on a perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. o 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. o Find two points on the perpendicular bisector 𝑥 +𝑥 𝑦 +𝑦 Find the midpoint between the two points- ( 1 2 2 , 1 2 2) 𝑦 −𝑦 Then find the slope between the two points 𝑥2−𝑥1 2 1 Find the slope that is perpendicular (opposite sign/reciprocal) to the slope that you just found. Use that to find additional points Angle Bisector o A ray that divides an angle into two congruent angles. o Angle Bisector Theorem- If a point is on the bisector of an angle then the point is equidistant from the sides of the angle. o Converse of the Angle Bisector 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. Section 3: Concurrent Lines, Medians, and Altitudes Point of concurrency o where three or more lines intersect Concurrent o when three or more lines intersect in one point. Perpendicular Bisectors o Theorem 5-6: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. o The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter. Angle Bisectors o Theorem 5-7: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides o The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle. Median o A segment whose endpoints are a vertex and the midpoint of the opposite side. o Theorem 5-8: 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. o The point of concurrency of the medians is the centroid Altitude o A perpendicular segment from vertex to the line containing the opposite side of a triangle. o Altitudes can be inside, on or outside the triangle. o Theorem 5-9: The lines that contain the altitudes of a triangle are concurrent o The point of concurrency of the altitudes is the orthocenter. Section 5: Inequalities in Triangles Corollary to the Triangle Exterior Angle Theorem-The measure of an exterior angles of a triangle is greater than the measure of each of its remote interior angles. Theorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. Theorem 5-11: If two angles of a triangle are not congruent, then the longest side lies opposite the larger angle. Smallest Angles opposite smallest sidebiggest angles opposite biggest sides Triangle Inequality Theorem- The sum of the lengths of any two sides of a triangle is greater than the length of the third side o Determine if the three sides can make a triangle o Determine the possible third side if given two triangles (answer must be written as a compound inequality) Review Problems Chapter Review o Page 297-299 #1-42 skipping any problems that pertain to sec. 4 Chapter Test o Page 300 #7-24 skipping any problems that pertain to sec. 4 Extra Practice o Page 724-725 #1-42 skipping any problems that pertain to sec. 4