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. Congruent
Triangles
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The Triangle Congruence Postulates &Theorems LA HALL HL FOR
RIGHT TRIANGLES ONLY AASASA SAS SSS FOR ALL TRIANGLES
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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. zThink 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. zBut do
the two triangles have to be congruent? 8530 8530
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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) zIf 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! F E D A
C B
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AAS (Angle, Angle, Side) Special case of ASA zIf 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! F E D A C B
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SAS (Side, Angle, Side) zIf 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! F E D A
C B
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SSS (Side, Side, Side) zIn two triangles, if 3 sides of one are
congruent to three sides of the other,... F E D A C B then the 2
triangles are CONGRUENT!
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Example 1 zGiven the markings on the diagram, is the pair of
triangles congruent by one of the congruency theorems in this
lesson? F E D A C B
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Example 2 zGiven the markings on the diagram, is the pair of
triangles congruent by one of the congruency theorems in this
lesson? A C B F E D
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Example 3 zGiven the markings on the diagram, is the pair of
triangles congruent by one of the congruency theorems in this
lesson? D A C B
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Example 4 Why are the two triangles congruent? What are the
corresponding vertices? A B C D E F SAS A D C E B F
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Example 5 Why are the two triangles congruent? What are the
corresponding vertices? A B C D SSS A C ADB CDB ABD CBD
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Example 6 Given: B C D A Are the triangles congruent? SSSSSS
Why?
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Example 7 Given: RHSRHS n Are the Triangles Congruent? QSR PRS
= 90 Q R S P T m QSR = m PRS = 90 Why?
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Summary: ASA - Pairs of congruent sides contained between two
congruent angles SAS - Pairs of congruent angles contained between
two congruent sides SSS - Three pairs of congruent sides AAS Pairs
of congruent angles and the side not contained between them.
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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