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(5.1) Midsegments of Triangles What will we be learning today? Use properties of midsegments to solve problems. Theorem 5-1: Triangle Midsegment 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 its length Key Terms: A midsegment of a triangle is connecting the midpoints of two sides. a segment Theorem 5-1: Triangle Midsegment 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 its length Example 1: Finding Lengths In XYZ, M, N and P are the midpoints. The Perimeter of and YZ. MNP is 60. Find NP Because the perimeter is 60, you can find NP. x NP + MN + MP = 60 (Definition of Perimeter) NP + NP + + = 60 24 M = 60 P 22 NP = Y N Z Theorem 5-1: Triangle Midsegment 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 its length Example 1: Use the Triangle Midsegment Theorem to find YZ MP = of YZ Triangle Midsegment Thm. MP = 24 24 = ½ YZ = YZ x Substitute 24 for MP Multiply both sides by 2 or the reciprocal of ½. 24 M P 22 Y N Z Theorem 5-1: Triangle Midsegment 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 its length Example 2: Identifying Parallel Segments Find the m<AMN and m<ANM. Line segments MN and BC are cut by transversal AB, so m<AMN and <B corresponding are angles. A Line Segments MN and BC are parallel by the Triangle Midsegment Theorem, so m<AMN is congruent to <B by the Corresponding Angles Postulate. m<AMN = 75 because congruent angles have the same measure. In triangle AMN, AM = AN , so m<ANM = m<AMN by the Isosceles Triangle Theorem. m<ANM = 75 by substitution. N M M 75O C B Theorem 5-1: Triangle Midsegment 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 its length Quick Check: 1. AB = 10 and CD = 28. Find EB, BC, and AC. A E D B C Theorem 5-1: Triangle Midsegment 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 its length Quick Check: 2. Critical Thinking Find the m<VUZ. Justify your answers. X 65O U Y V Z HOMEWORK (5.4) Pgs. 325-363; 18- 26,27,49 (5.2) Bisectors in Triangles What will we be learning today? Use properties of perpendicular bisectors and angle bisectors. Theorems Theorem 5-2: Perpendicular Bisector Thm. Theorem 5-3: Converse of the Perpendicular Bisector Thm. If a point is on the perpendicular bisector of a segment, then it is equidistant form the endpoints of the segment. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. Theorems Theorem 5-4: Angle Bisector Thm. Theorem 5-5: Converse of the Angle Bisector Thm. If a point is on the bisector of an angle, then it is If a point in the interior of an equidistant from the sides angle is equidistant from the of the angle. sides of the angle, then it is on the angle bisector. Key Concepts The distance from a point to a line is the length of the perpendicular segment from the point to the line. Example: D is 3 in. from line AB and line AC C D 3 A B Example Using the Angle Bisector Thm. Find x, FB and FD in the diagram at the right. Show steps to find x, FB and FD: FD = 7x – 35 = 2x + 5 A 2x + 5 B F C Angle Bisector Thm. D 7x - 35 E Quick Check a. According to the diagram, how far is K from ray EH? From ray ED? 2xO D E C (X + 20)O K 10 H Quick Check b. What can you conclude about ray EK? 2xO D E C (X + 20)O K 10 H Quick Check c. Find the value of x. 2xO D E C (X + 20)O K 10 H Quick Check d. Find m<DEH. 2xO D E C (X + 20)O K 10 H HOMEWORK (5.2) Pgs. 267-269; 1-4, 6, 8-26, 28, 29, 40, 43, 46, 48