Traditional Congruence Revisited

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Traditional Congruence Revisited

• Francisco Nolasco
• Math 5370d
• Dr. Winsor
• July 3, 2006

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Overview

• Review definitions
• Congruence of segments and angles
• Congruence of triangles
• SsA Theorem
• Problems 5,6,7,8

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Review Definitions

• Open http//www.mrnolasco.com/mat to get Word
  document with definitions
• Translation
• Synthetic
• Analytic
• Rotation
• Reflection
• Congruence Transformations

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Congruence of segments and angles

• Theorem 7.35 If two segments have the same
  length, then they are congruent.
• Proof using a translation and a rotation
• Proof using only reflections
• Corollary Two segments have the same length if
  and only if they are congruent.

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Congruence of segments and angles

• Theorem 7.36 If two angles have the same
  measure, then they are congruent.
• HW Proof using a translation and a rotation
• HW Proof using only reflections
• Corollary Two angles have the same measure if
  and only if they are congruent.

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Congruence of triangles

• Theorem 7.37 (SAS Congruence) If two sides and
  the included angle of one triangle are congruent
  to two sides and the included angle of a second
  triangle, then the triangles are congruent.
• Proof using a translation and a rotation
• Proof using only reflections
• Proof using angle congruence

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Congruence of triangles

• Theorem 7.38 (ASA Congruence) If two angles and
  the included side of one triangle are congruent
  respectively to two angles and the included side
  of another triangle, then the triangles are
  congruent.
• HW Proof using a translation and a rotation
• HW Proof using only reflections
• HW Proof using angle congruence

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Congruence of triangles

• Theorem 7.39 (SSS Congruence) If the three sides
  of one triangle are congruent to the three sides
  of another triangle, then the triangles are
  congruent.
• Proof using kites

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Congruence of triangles

• Theorem 7.40 (SsA Congruence) If two sides and
  the angle opposite the longer of the two sides in
  one triangle are congruent, respectively, to two
  sides and the corresponding angle in another
  triangle, then the triangles are congruent.
• Corollary (HL Congruence) If the hypotenuse and
  leg of one right triangle are congruent to the
  hypotenuse and leg of another right triangle,
  then the triangles are congruent.

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Class Problems
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Class Problems
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Class Problems
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Class Problems
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Class Problems
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Conclusion

• It is not often proved that figures need to
  overlap to be congruent.
• Our students may benefit from the demonstrations
  of the proofs in this section.