Properties of Triangles

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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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