Geometry Using Triangle Congruence and Similarity

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Chapter 14

• Geometry Using Triangle Congruence and Similarity

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14.1 Congruence of Triangles

• If two lines are of the same length, or if two
  angles have the same measure, we call them
  congruent lines or congruent angles, respectively.

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Suppose we have two triangles, ?ABC and ?DEF, and
that we pair up the vertices, A?D, B?E and C?F.
In this fashion, we have formed a correspondence
between ?ABC and ?DEF.
Side AB corresponds to side DE, BC corresponds to
EF, and CA corresponds to FD.
Similarly, angle CAB corresponds to angle FDE,
etc.
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• Definition Suppose that ?ABC and ?DEF are such
  that under the correspondence A?D, B?E and C?F
  all corresponding sides are congruent and all
  corresponding vertex angles are congruent. Then
  ?ABC is congruent to ?DEF and we write

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• Property Side-Angle-Side (SAS) Congruence
• 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.

Property Angle-Side-Angle (ASA) Congruence 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
triangles are congruent.
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• Theorem
• Opposite sides of a parallelogram are congruent.
• Opposite angles of a parallelogram are congruent.

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Property Side-Side-Side (SSS) Congruence If
three sides of a triangle are congruent,
respectively, to three sides of another triangle,
then the triangles are congruent.
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14.2 Similarity of Triangles

• Two geometric figures that have the same shape
  but are not necessarily the same size are called
  similar.

Definition Suppose that ?ABC and ?DEF are such
that under the correspondence A?D, B?E and C?F
all corresponding sides are proportional and all
corresponding vertex angles are congruent. Then
?ABC is similar to ?DEF and we write
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• Similarity Properties of Triangles
• Two triangles, ?ABC and ?DEF, are similar if and
  only if at least one of the following three
  statements is true.
• Two pairs of corresponding sides are proportional
  and their 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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14.3 Basic Euclidean Constructions

• Compass and Straightedge Properties
• For every positive number r and for every point
  C, a circle of radius r and center C can be
  constructed. A connected portion of a circle is
  called an arc.
• Ever pair of points can be connected by our
  straightedge to construct a line, a segment or a
  ray.
• A line l can be constructed if and only if we
  have located two points that are on l.

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Constructions

1. Copy a line segment.
2. Copy an angle.
3. Construct a perpendicular bisector of a line
   segment.
4. Bisect an angle.
5. Construct a line perpendicular to a given line
   through a specified point on the line.
6. Construct a line perpendicular to a given line
   through a point not on the line.
7. Construct a line parallel to a given line through
   a specified point not on the line.

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14.4 Additional Euclidean Constructions

• 8. Construct the circumscribed circle of a
  triangle.
• 9. Construct the inscribed circle of a triangle.
• 10. Construct an equilateral triangle.

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Fermat Primes and Regular Polygon Constructions

• Definition A Fermat prime is a prime number of
  the form

Gauss Theorem for Constructible Regular n-gon A
regular n-gon can be constructed with a
straightedge and a compass if and only if the
only odd prime factors of n are distinct Fermat
Primes.