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(5.1) Midsegments of Triangles

What will we be learning today? Use properties of midsegments to solve problems.

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Key Terms: A midsegment of a triangle is

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 a segment connecting the midpoints of two sides.

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Example 1: Finding Lengths

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 MNP is 60. Find NP and YZ. Because the perimeter is 60, you can find NP. NP + MN + MP = 60 (Definition of Perimeter) NP = 60 NP + = 60 NP = x 24 M P 22 Y Z N

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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 = = ½ YZ Substitute 24 for MP = YZ Multiply both sides by or the reciprocal of ½. x 24 M P 22 Y Z N

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Example 2: Identifying Parallel Segments

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 are angles. Line Segments MN and BC are parallel by the Theorem, so m<AMN is congruent to <B by the Postulate. m<AMN = 75 because congruent angles have the same measure. In triangle AMN, AM = , so m<ANM = by the Triangle Theorem. m<ANM = by substitution. A corresponding Triangle Midsegment N M M Corresponding Angles AN m<AMN Isosceles 75O C 75 B

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AB = 10 and CD = 28. Find EB, BC, and AC.

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: AB = 10 and CD = 28. Find EB, BC, and AC. A E B C D

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2. Critical Thinking Find the m<VUZ. Justify your answers.

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 Z Y V

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HOMEWORK (5.1) Pgs ; 1, 4, 6, 7-11, 13, 14, 18, 20-22, 26, 34, 36

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(5.2) Bisectors in Triangles

What will we be learning today? Use properties of perpendicular bisectors and angle bisectors.

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Theorems Theorem 5-2: Perpendicular Bisector Thm. If a point is on the perpendicular bisector of a segment, then it is equidistant form the endpoints of the segment. Theorem 5-3: Converse of the Perpendicular Bisector Thm. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

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Theorems Theorem 5-4: Angle Bisector Thm. If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Theorem 5-5: Converse of the Angle Bisector Thm. If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the angle bisector.

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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

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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 = Angle Bisector Thm. 7x – 35 = 2x + 5 A 2x + 5 B F 7x – 35 C D E

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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

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Quick Check b. What can you conclude about ray EK? 2xO D E C (X + 20)O

10 H

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Quick Check c. Find the value of x. 2xO D E C (X + 20)O K 10 H

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Quick Check d. Find m<DEH. 2xO D E C (X + 20)O K 10 H

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HOMEWORK (5.2) Pgs ; 1-4, 6, 8-26, 28, 29, 40, 43, 46, 48

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