Geometry Lesson 1: Triangles

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Geometry Lesson 1 Triangles
Grade 9 Honors Geometry

• By Dusan Vidovic

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Purpose of the Lesson

• Addressing 4th Indiana Academic Standard for
  Geometry.
• Identify and describe triangles.
• Define, identify, and construct altitudes,
  medians, and angle bisectors.
• Identify congruent triangles.

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But First, What Is a Triangle?

• A polygon with 3 sides which are straight
  line segments.
• Base
• 2 legs
• Any 3 non-collinear points determine a
  triangle.
• Vertices
• 3 angles summing up to 180 degrees.

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Triangle Classification

• According to the lengths of their sides.
• Isosceles
• Scalene
• Equilateral
• Equiangular

• According to the size of their largest internal
  angle.
• Right
• Obtuse
• Acute

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Isosceles Triangle

• At least two sides are of equal length.
• It also has two congruent angles.

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Scalene Triangle

• All sides have different lengths.
• The internal angles are all different.

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Equilateral/Equiangular Triangle

• All sides are of equal length.
• All its internal angles are equal.
• An equilateral triangle is also equiangular.

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Right Triangle

• It has one 90 internal angle (a right angle).
• The side opposite to the right angle is the
  hypotenuse (the longest side in the right
  triangle).
• The other two sides are the legs.
• Opposite
• Adjacent

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Obtuse Triangle

• It has one internal angle larger than 90.

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Acute Triangle

• It has internal angles that are all smaller than
  90 (three acute angles).

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Triangle Properties

• All of the previous triangles that we have seen
  have the following properties
• Medians
• Altitudes
• Angle Bisectors
• The hyperlink on each term will lead you to more
  information including formulas which are not
  covered in today's lecture.

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Median

• It is a line joining a vertex on one angle to the
  midpoint of the opposite side.
• It divides the triangle into two parts of equal
  area.
• Every triangle has three medians which intersect
  in the triangle's center of mass.

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Altitude

• It is a straight line through a vertex and
  perpendicular to (i.e. forming a right angle
  with) the opposite side.
• The intersection between the (extended) side and
  the altitude is called the foot of the altitude.
  This opposite side is called the base of the
  altitude.
• The length of the altitude is the distance
  between the base and the vertex.

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Angle Bisector

• The (interior) bisector of an angle is the line
  segment that divides the angle into two equal
  parts.

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Triangle Congruence

• Two triangles are congruent if their
  corresponding sides and angles are equal.
• Usually it is sufficient to establish the
  equality of three corresponding parts and use one
  of the following results to conclude the
  congruence of the two triangles.
• SAS - SSS ASA AAS

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Side Angle Side

• Two triangles are congruent if a pair of
  corresponding sides and the included angle are
  equal.

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Side Side - Side

• Two triangles are congruent if their
  corresponding sides are equal.

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Angle Side - Angle

• Two triangles are congruent if a pair of
  corresponding angles and the included side are
  equal.

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Angle Angle - Side

• Two triangles are congruent if two angles and a
  non - included side
• of one triangle are congruent to two angles and
  the corresponding
• non-included side of another triangle.

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Important Remarks

• An equilateral triangle is also an isosceles
  triangle, but not all isosceles triangles are
  equilateral triangles.
• An equilateral triangle is an acute triangle, but
  not all acute triangles are equilateral
  triangles.
• AAA (Angle-Angle-Angle) is not a method of
  proving triangle congruence because it says
  nothing about the size of the two triangles and
  hence shows only similarity and not congruence.
• While the AAS (Angle-Angle-Side) condition
  guarantees congruence, SSA (Side-Side-Angle) does
  not. WHY?????????
• More information on triangles can be found at
  Math World.

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Sources

• http//mathworld.wolfram.com/Triangle.html
• http//www.indianastandards.org/standard.asp?Subje
  ctmathGradeGStandard4
• http//library.thinkquest.org/20991/geo/ctri.html
  SAS
• Pictures
• http//ostermiller.org/calc/triangle.png
• http//academics.sru.edu/ModLang/French/Activities
  /french20photo20gallery/paris20-20rose/glass2
  0pyramid20-20louvre.jpg
• http//www.pbs.org/wgbh/nova/pyramid/geometry/imag
  es/scaleintro.jpeg
• http//en.wikipedia.org/wiki/ImageTriangle.Equila
  teral.svg
• http//en.wikipedia.org/wiki/ImageTriangle.Isosce
  les.svg
• http//en.wikipedia.org/wiki/ImageTriangle.Scalen
  e.svg
• http//en.wikipedia.org/wiki/ImageTriangle.Right.
  svg
• http//en.wikipedia.org/wiki/ImageTriangle.Obtuse
  .svg
• http//en.wikipedia.org/wiki/ImageTriangle.Acute.
  svg
• http//en.wikipedia.org/wiki/ImageTriangle.Centro
  id.svg
• http//en.wikipedia.org/wiki/ImageTriangle.Orthoc
  enter.svg
• http//library.thinkquest.org/20991/geo/ctri.html
  SAS

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THE END

• Homework
• Study the material presented today.
• Do Pg. 49, numbers 3,5,7,9,11,12, and 14 for
  Monday.
• Have a great weekend!!!