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Chapter 3: Polygons and Their Relations

   Here are some additional theorems about the congruence of triangles:

  Corresponding medians of congruent triangles are congruent.

  Corresponding altitudes of congruent triangles are congruent.

  The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

  The bisector of the vertex angle of an isosceles triangle is also a median and altitude of the triangle.

  Line containing two points, each equidistant from the endpoints of a given segment, is the perpendicular bisector of the segment.

  Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

    If the triangles can be congruent, then they can also be non-congruent or not equal in size. There are many ways to prove that two triangles are not congruent. The most obvious way is that all three sides or angles are different. Here are some additional ways to prove that two triangles are not congruent:

  Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

  The Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

  The SAS Inequality Theorem: If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and the included angle of the first triangle has a greater measure than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

  The SSS Inequality Theorem: If two sides of one triangle are congruent, respectively, to two sides of a second triangle, and the length of the third side of the first triangle is greater than the length of the third side of the second triangle, then the angle opposite the third side of the first triangle ahs a greater measure than the angle opposite the third side of the second triangle.

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