Geometry

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Geometry
Triangle Congruence Theorems
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Congruent Triangles

• Congruent triangles have three congruent sides
  and and three congruent angles.
• However, triangles can be proved congruent
  without showing 3 pairs of congruent sides and
  angles.

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The Triangle Congruence Postulates Theorems
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Theorem

• If two angles in one triangle are congruent to
  two angles in another triangle, the third angles
  must also be congruent.

• Think about it they have to add up to 180.

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A closer look...

• If two triangles have two pairs of angles
  congruent, then their third pair of angles is
  congruent.

• But do the two triangles have to be congruent?

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Example
Why arent these triangles congruent? What do
we call these triangles?
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• So, how do we prove that two triangles really are
  congruent?

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ASA (Angle, Side, Angle)

• If two angles and the included side of one
  triangle are congruent to two angles and the
  included side of another triangle, . . .

then the 2 triangles are CONGRUENT!
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AAS (Angle, Angle, Side)Special case of ASA

• 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, . . .

then the 2 triangles are CONGRUENT!
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SAS (Side, Angle, Side)

• If in two triangles, two sides and the included
  angle of one are congruent to two sides and the
  included angle of the other, . . .

then the 2 triangles are CONGRUENT!
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SSS (Side, Side, Side)

• In two triangles, if 3 sides of one are congruent
  to three sides of the other, . . .

then the 2 triangles are CONGRUENT!
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HL (Hypotenuse, Leg)

• If both hypotenuses and a pair of legs of two
  RIGHT triangles are congruent, . . .

then the 2 triangles are CONGRUENT!
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HA (Hypotenuse, Angle)

• If both hypotenuses and a pair of acute angles of
  two RIGHT triangles are congruent, . . .

then the 2 triangles are CONGRUENT!
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LA (Leg, Angle)

• If both hypotenuses and a pair of acute angles of
  two RIGHT triangles are congruent, . . .

then the 2 triangles are CONGRUENT!
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LL (Leg, Leg)

• If both pair of legs of two RIGHT triangles are
  congruent, . . .

then the 2 triangles are CONGRUENT!
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Example 1

• Given the markings on the diagram, is the pair of
  triangles congruent by one of the congruency
  theorems in this lesson?

D
E
F
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Example 2

• Given the markings on the diagram, is the pair of
  triangles congruent by one of the congruency
  theorems in this lesson?

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Example 3

• Given the markings on the diagram, is the pair of
  triangles congruent by one of the congruency
  theorems in this lesson?

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Example 4

• Why are the two triangles congruent?
• What are the corresponding vertices?

SAS
?A ? ? D
?C ? ? E
?B ? ? F
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Example 5
A

• Why are the two triangles congruent?
• What are the corresponding vertices?

SSS
B
D
?A ? ? C
?ADB ? ? CDB
C
?ABD ? ? CBD
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Example 6

• Given

Are the triangles congruent?
Why?
S S S
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Example 7

• Given

m?QSR m?PRS 90

• Are the Triangles Congruent?

Why?
R H S
?QSR ? ?PRS 90
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Summary
ASA - Pairs of congruent sides contained between
two congruent angles
AAS Pairs of congruent angles and the side not
contained between them.
SAS - Pairs of congruent angles contained between
two congruent sides
SSS - Three pairs of congruent sides
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Summary ---for Right Triangles Only
HL Pair of sides including the Hypotenuse and
one Leg HA Pair of hypotenuses and one acute
angle LL Both pair of legs LA One pair of
legs and one pair of acute angles
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THE END!!!