MODULE 3

• Discipline


  Good taste<|>Excellence
• In this module you will learn how to:
• Lesson 1

  Proves the Midline Theorem
• Lesson 2


  Proves theorems on trapezoids and kites
• Midline Theorem


  The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long
• Midsegment of a trapezoid


  The line segment connecting the midpoints of the nonparallel sides
• The three midsegments of a triangle divide the triangle into 4 congruent triangles
• Midsegment of a triangle


  • In triangle MCG, A and I are the midpoints of MG and GC respectively
• Midsegment length


  Equal to half the length of the third side
• If the midsegment length is 10.5 units, then the length of the third side is 21 units
• If the length of the third side is 32 units, then the midsegment length is 16 units
• Congruence

  The line segment connecting the midpoints of the nonparallel sides of a trapezoid is called the midsegment
• Triangle Midsegment Theorem


  A midsegment of a triangle is parallel to the third side and is equal to half the length of the third side
• Corresponding angles of parallel lines cut by a transversal

  Are congruent
• Proof of congruence of triangles

  Shown using SAS (Side-Angle-Side)
• The same argument can be used to prove the congruence of the other two triangles KML and NKM
• Given information

  AI = 10.5 units
• AI = 1/2 of MC


  By Triangle Midsegment Theorem
• Given information

  CG = 32 units
• GI = 1/2 of CG


  By Triangle Midsegment Theorem
• Given information


  AI = 3x - 2, MC = 9x - 13
• Solving for x, AI, and MC
• MG is congruent to CG
• AG = 1/2 of MG


  By Triangle Midsegment Theorem
• IC = 1/2 of GC


  By Triangle Midsegment Theorem
• AG is congruent to IC
• MG = 1/2 of AG


  By Triangle Midsegment Theorem
• GC = 1/2 of IC


  By Triangle Midsegment Theorem
• Solving for MG and CG
• Trapezoid Midsegment Theorem


  A midsegment of a trapezoid is parallel to the bases and is equal to half of the sum of the lengths of the bases
• Given information

  MD = 22, CE = 36
• MD = 1/2 of (UT + CE)


  By Trapezoid Midsegment Theorem
• UT = 8 units
• Given information


  UM = CM, TD = ET, UT = 10, CE = 23, DE = 7, UM = 9
• Solving for MD and the perimeter of trapezoid CUTE
• Two-column proof


  Prove that TR = 1/2 (MS + IN)
• Given: TR || IN, TR || MS
• Proof steps

  Given

  2. Draw IS with P as its midpoint

  3. TP = 1/2 MS, TP || MS (by Triangle Midsegment Theorem)

  4. MS || IN

  5. T, P, and R are collinear

  6. TR = TP + PR

  7. TR = 1/2 (MS + IN)
• 1. 2-1 Go! - 3 things/facts learned, 2 questions, 1 opinion
• Performance Task - Create a scrapbook on Midline Theorem