Congruence of Triangles

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Congruence of Triangles

• (NCTM standard) All grades 3-5 should explore
  congruence and similarity of shapes.
• Notation
• stands for segment AB,
• stands for ray from A to B (A is the fixed
  point)

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• represent the angle ABC
• represents the triangle ABC
• If two line segments AB and CD are equal in
  length we say segment AB is congruent to segment
  CD and write it as .
• if two angles and are
  equal in measure we say the two angles are
  congruent and write it as

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• Two triangles and are said to
  be congruent if and only if there is a one to one
  correspondence between the respective sides and
  angles such that the corresponding angles and
  sides are congruent. We denote this by
  .

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• Properties of Congruence
• (SAS) Side Angle Side - If two sides and the
  included angle of a triangle are congruent,
  respectively, to two sides and the included angle
  of another triangle, then the triangles are
  congruent.

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• (ASA) Angle Side Angle - If two angles and the
  included side of a triangle are congruent,
  respectively, to two angles and the included side
  of another triangle, then the two triangles are
  congruent.

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• (SSS) Side-Side-Side - If three sides of a
  triangle are congruent, respectively to three
  sides of another triangle, then the two triangles
  are congruent.

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Examples

• Show that the triangles below are congruent to
  each other.
• 1.

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• (NCTM standard- grades 6-8 learn deductive and
  inductive arguments concerning geometric ideas
  and relationships, such as congruence and
  similarity, and the Pythagorean relationship.
• Make students recognize that reasoning and proof
  are essential and powerful parts of mathematics.

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Similarity of Triangles

• Two triangles and are
  said to be similar if and only if all
  corresponding sides are proportional and all
  corresponding vertex angles are congruent. We
  denote this by

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• Properties of Similar triangles-
• Two pairs of corresponding sides are
  proportional and their included included angles
  are congruent (SAS similarity)
• Two pairs of corresponding angles are congruent
  (AA similarity)
• All three pairs of corresponding sides are
  proportional. (SSS similarity)

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• http//animal.discovery.com/games/shapeshifter/sha
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