Isosceles and Equilateral Triangles

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Isosceles and Equilateral Triangles

• Academic Geometry

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Isosceles and Equilateral Triangles

• Draw a large isosceles triangle ABC, with exactly
  two congruent sides, AB and AC.
• What is symmetry?
• How many lines of symmetry does it have?
• Label the point of intersection D.

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Isosceles and Equilateral Triangles

• What is the relationship between AD and BC?

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Isosceles and Equilateral Triangles

• Draw a Triangle XYX with exactly two congruent
  angles, ltY and ltZ. Find the line of symmetry.
• What can you conclude about the sides?

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Isosceles Triangle Theorems

• The congruent sides of an isosceles trianlge are
  its legs.
• The third side is the base.
• The two congruent sides form the vertex angle.
• The other two angles are base angles.

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Theorem 4-3

• Isosceles Triangle Theorem
• The base angles of an isosceles triangle are
  congruent.
• If the two sides of a triangle are congruent,
  then the angles opposite those sides are
  congruent.

c
a
b
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Theorem 4-4

• Converse of Isosceles Triangle Theorem
• If two angles of a triangle are congruent, then
  the sides opposite those angles are congruent.

c
b
a
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Theorem 4-5

• The line of symmetry for an isosceles triangle
  bisects the vertex angle and is the perpendicular
  bisector of the base.
• CD AB
• CD bisects AB

c
a
b
d
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Example

• EC is a line of symmetry for isosceles triangle
  MCJ.
• Draw and label the triangle.
• MltMCJ 72. Find mltMEC, mltCEM and EJ.
• ME 3

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Proof of the Isosceles Triangle Theorem

• Begin with isosceles triangle XYZ. XY is
  congruent XZ. Draw XB, the bisector of the
  vertex angle YXZ
• Prove ltY congruent ltZ
• Statements Reasons

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Using the Isosceles Triangle Theorems

• Why is each statement true?
• ltWVS congruent ltS
• TR congruent TS
• Can you deduce that Triangle RUV is isosceles?
  Explain

t
u
w
r
s
v
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Using Algebra

• Find the value of y

m
y
63
l
n
o
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Equilateral Triangles

• Draw a large equilateral triangle, EFG.
• Find all the lines of symmetry. How many are
  there?
• What do we know about the sides?
• The angles?

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Isosceles and Equilateral Triangles

• We learned in the last chapter that equilateral
  triangles are also isosceles.
• A corollary is a statement that immediate follows
  from a theorem.

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Corollary to Theorem 4-3

• If a triangle is equilateral, then the triangle
  is equiangular.
• ltX is congruent to ltY is congruent to ltZ

y
z
x
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Corollary to Theorem 4-4

• If the triangle is equiangular, then the triangle
  is equilateral.
• XY is congruent to YZ is congruent to ZX

y
z
x