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Inequalities and Triangles Sec: 5.6 Sol:G.5 Theorem 5.1 Mid-segment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half the length of that side. B DE AC D A E C 1 DE AC 2 Theorem 5.8 Exterior angle inequality theorem: If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles. B Ex: m4 > m2 m4 > m1 2 1 A 4 3 C Theorem 5.9 Side – Angles Relationship If one side of the triangle is longer than another side, then the angle opposite the longer side has a greater measure then the angle opposite the shorter side. P O R Ex: Determine the relationship between the measures of the given angles. R 5.2 U a) RSU , SUR b) TSV , STV c) RSV , RUV 6.6 S 5.1 V T EX: Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds it with the dimensions shown, which two ends should she tie? Y 22.6 X Z Theorem 5.10 Angle – Side Relationship • If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. P R Q Section 5.4 Theorem 5.11 Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A All of the following must be true: AB+BC>AC BC+AC>AB B AC+AB>BC C *Theorem 5.11 is used to determine whether the three given segments can form a triangle.* Examples: 1. 2. 3. 4. 2,4,5 6.8, 7.2, 5.1 6.5, 6.5, 14.5 In Triangle PQR, PQ=7.2,and QR=5.2 Which measure cannot be PR? a) 7 b) 9 c) 11 d) 13 Find the range for the measure of the third side. 1. 6 and 7 2. 5 and 2 3. 7 and 17 Distance Between a point and a line: Theorem 5.12 : The perpendicular segment from a point to a line is the shortest segment from the point to the line. P EX: PQ is the shortest segment from P to AB A Q B Corollary 5.1: The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. A Which segment is the shortest possible distance from point A to Plane P? B F E D C P Suggested assignment Classwork: Workbook pg 139-140 all Homework: Pg 329 10-32 even