Geometry

GeometryTriangle Congruence Theorems

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

The Triangle Congruence Postulates &Theorems

LAHALLHLFOR RIGHT TRIANGLES ONLY

AASASASAS

SSSFOR ALL TRIANGLES

TheoremIf 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.

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?8530

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Example

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Why arent these triangles congruent? What do we call these triangles?

So, how do we prove that two triangles really are congruent?

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!

FED

ACB

AAS (Angle, Angle, Side)Special case of ASAIf 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!

FED

ACB

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!

FED

ACB

SSS (Side, Side, Side)In two triangles, if 3 sides of one are congruent to three sides of the other, . . .

FED

ACB

then the 2 triangles are CONGRUENT!

HL (Hypotenuse, Leg)If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . .

ACB

FED

then the 2 triangles are CONGRUENT!

HA (Hypotenuse, Angle)If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . .then the 2 triangles are CONGRUENT!FED

ACB

LA (Leg, Angle)If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . .then the 2 triangles are CONGRUENT!

ACB

FED

LL (Leg, Leg)If both pair of legs of two RIGHT triangles are congruent, . . .then the 2 triangles are CONGRUENT!

ACB

FED

Example 1Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?FED

ACB

Example 2Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

ACB

FED

Example 3Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson?

D

ACB

Example 4Why are the two triangles congruent?

What are the corresponding vertices?ABCDEF

SASA DC EB F

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Example 5Why are the two triangles congruent?

What are the corresponding vertices?

ABCD

SSSA CADB CDBABD CBD

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Example 6Given:

BCDA

Are the triangles congruent?

SSSWhy?

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Example 7Given:

RHS

Are the Triangles Congruent? QSR PRS = 90

QRSPT

mQSR = mPRS = 90

Why?

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Summary:ASA - Pairs of congruent sides contained between two congruent anglesSAS - Pairs of congruent angles contained between two congruent sidesSSS - Three pairs of congruent sidesAAS Pairs of congruent angles and the side not contained between them.

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

THE END!!!