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Chapter 5: Properties of Triangles Geometry Fall 2008 5.1 Perpendiculars and Bisectors Perpendicular Bisector: a segment, ray, or line that is perpendicular to a segment at its midpoint. Equidistant: the same distance from Perpendicular Bisector Theorem Bisector Theorem: If a point is on the bisector of a segment, then it is equidistant from the endpoints of the segment. P If PM bisects AB and is to AB, then A M B AP = BP. Converse If AP = BP, then P is on the bisector of AB. Angle Bisector Theorem If a point is on the bisector of an angle then it is equidistant from the two sides of an angle. B D If m BAD=m CAD A then DB = DC. C Converse: If DB = DC then m BAD = m CAD 5.2 Bisectors of a Triangle Perpendicular bisector of a triangle: line, ray, or segment that is perpendicular to a side of the triangle at the midpoint of the side. Concurrent Lines When 3 or more lines intersect in the same point they are called concurrent lines. Then intersection is called the concurrency. The 3 perpendicular bisectors of a triangle are concurrent. Acute, Right and Obtuse Inside On Outside Circumcenter The intersection of the three perpendicular bisectors is called the circumcenter. The circumcenter is equidistant from each of the vertices. Angle Bisector of a Triangle Angle bisector of a triangle: a line, segment, or ray that bisects and angle of the triangle Incenter The 3 angle bisectors are concurrent. The intersection of the 3 angle bisectors is called the incenter. Incenter The incenter is always inside the triangle and is equidistant from the sides of the triangle. GB = GD = GF D F G B Is C on the perpendicular bisector of AB? A B C Is C on the perpendicular bisector of AB? C A B Is P on the bisector of A? P A Is P on the bisector of A? 4 A P 4 5.2 examples B Find the length of DA. Find the length of AB. E D Why is ADF = BDE? 2 A C F Examples V is the incenter of XWZ. Length of VS: m VZX = W S T 3 V 20 Z X Y Find ID C 8 10 D I A B Find BD C D 15 A B 12 12 Homework p. 268 # 8-13, 16-26 p. 275 # 10-17 5.3 Medians and Altitudes of a Triangle Median of a Triangle: a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Centroid The three medians are concurrent. The intersection of the three medians is called the centroid of the triangle. Concurrency of Medians of a triangle The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. B If P is the centroid of ABC, then D AP = 2/3 AD, C BP = 2/3 BF, E P and CP = 2/3 CE. F A Altitudes of a Triangle The altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. The intersection is called the orthocenter. BAE = EAC and BF = FC Median: Altitude: A Bisector: Bisector: a) AD b) AE G c) AF B D E F C d) GF C is the centroid of XYZ. YI = 9.6 X CK = XK = YC = 8 J I KZ = C JZ = Perimeter of : Y K Z 5 5.4 Midsegment Theorem A midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle. Midsegment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. Example 1 D, E, and F are the midpoints of the sides. A DE ll _____ FE ll _____ If AB = 14, D F then EF = _____ If DE = 6, B C then AC = _____ E Example 2 R, S, and T are the midpoints of the sides. RK = 3, KS = 4, and JK KL J RS = _____ JK = _____ RT = _____ R T Perimeter = _____ 3 K 4 S L Example 3 If YZ = 3x + 1, and MN = 10x – 6, then YZ = ______ M X Z O Y N Example 4 If m MON = 48 , then m MZX = _____ M X Z O Y N Example 5 Find the midpoint of JK. Find the length of NK. Find the length of NP. Find the coordinates of point P. Homework for 5.3-5.4 P. 282 #8-11, 17-20 P. 290 # 12-18, 28-29 5.5 Inequalities in One Triangle If side 1 of a triangle is larger than side 2, then the angle opposite side 1 is larger than the angle opposite side 2. B Since BC > AB, A > C 5 3 A C Converse If angle 1 is larger than angle 2, then the side opposite angle 1 is longer than the side opposite angle 2. B Since A > C, BC > AB 60 40 A C Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. A 1 B C m 1 > m A and m 1 > m B Triangle Inequality The sum of the lengths of any 2 sides of a triangle is greater than the length of the 3rd side AB + BC > AC A AC + BC > AB AB + AC > BC C B Example 1 Name the shortest and longest sides. Q 65 35 P R Example 2 Name the smallest and largest angles. Z 4 X 6 8 Y Example 3 List the sides in order from shortest to longest. B C 95 30 A Example 4 Find the possible measures for XY in XYZ. XZ = 2 and YZ = 3 Example 5 Determine whether the given lengths could represent the lengths of the sides of a triangle: A) { 5, 5, 8} B) { 5, 6, 7} C) { 1, 2, 5} D) { 7, 3, 9} Homework 5.5 Pg. 298 # 6-11, 14-19, 24-25 5.6 Indirect Proof and Inequalities in Two Triangles Indirect Proof: a proof in which you prove that a statement is true by first assuming that its opposite is true. Writing an indirect proof Assume that the negation of the conclusion (what you are trying to prove) is true. Show that the assumption leads to a contradiction of known facts or of the given information. Conclude that since the assumption is false the original conclusion is true. Example 1 Given: ABC Prove: ABC does not have more than one obtuse angle. Begin by assuming that ABC does have more than one obtuse angle. m A > 90 and m B > 90 m A + m B > 180 Example 1 You know that the sum of the measures of all three angles is 180 . m A + m B + m C = 180 m A + m B = 180 - m C You can substitute 180 - m C for m A + m B in m A + m B > 180 . 180 - m C > 180 0 >m C Example 1 The last statement is “not possible.” Angle measure in triangles cannot be negative. So you can conclude that the original assumption must be false. That is, ABC cannot have more than one obtuse angle. Hinge Theorem If 2 sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. R V 100 80 S T X W Converse of Hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. R V 10 8 S T X W Example 1 Complete with <, >, or =. AB ____ DE C 105 B E A D 110 F Example 2 Complete with <, >, or =. m 1 ____ m 2 2 8 7 1 Homework 5.6 Pg. 305 # 3-5, 6-20

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