Angle Side Theorems 3'7

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Angle Side Theorems 3.7
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Big side ? Big angle Big angle ? Big side

• ..\..\..\..\Geometry\Heath geometry\HChpt
  5-Triangle Prop\TRNGLPTS\5_3big angle big side.gsp

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Big side ? Big angle

• Theorem
• If one side of a triangle is longer than another
  side, then the angle opposite the longer side is
  larger than the angle opposite the shorter side.

Longer side Larger angle
Shorter side Smaller angle
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Theorem

• If one side of a triangle is longer than another
  side, then the angle opposite the longer side has
  a greater measure than the angle opposite the
  shorter side.

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Big angle ?Big side

• Theorem
• If one angle of a triangle is larger than
  another angle, then the side opposite the larger
  angle is longer than the side opposite the
  smaller angle.

Longer side Larger angle
Shorter side Smaller angle
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Theorem

• If one angle of a triangle has a greater measure
  than another angle, then the side opposite the
  greater angle is longer than the side opposite
  the lesser angle.

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Corollary

• Corollary The perpendicular segment from a
  point to a line is the shortest segment that can
  be drawn from a point to the line.

o
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Theorem

• If one side of a triangle is longer than another
  side, then the angle opposite the longer side has
  a greater measure than the angle opposite the
  shorter side.

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Theorem

• If one angle of a triangle has a greater measure
  than another angle, then the side opposite the
  greater angle is longer than the side opposite
  the lesser angle.

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Isosceles Triangle Parts
A
Base
?Legs
? Base ?
B
C
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Isosceles Triangles

• If two side of a triangle are congruent, the
  angles opposite the side are congruent.
• If two angles of a triangle are congruent, the
  sides opposite the angles are congruent.
• If two side of a triangle are not congruent, the
  angles opposite the side are not congruent.
• If two angles of a triangle are not congruent,
  the sides opposite the angles are not congruent.

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Isosceles Triangle Theorem
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Corollaries

• A triangle is equilateral if and only if it is
  equiangular.
• Each angle of an equilateral triangle measures 60
  degrees.

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Isosceles Triangle Converse

• If two angles of a triangle are congruent, then
  the sides opposite those angles are congruent.
• Summary - If you have two congruent angles, then
  you have two congruent legs.

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Proof
Given BC?AD , lt1 ? lt2, and CF?AF .Prove ?CEF ?
?AEF.
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Proof
Given BC?DA , lt1 ? lt2, and CF?AF .Prove ?CEF ?
?AEF.
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Name the Congruent s
?ADC ? ?FBC