Geometry Triangle Congruence Theorems

zCongruent triangles have three congruent sides and and three congruent angles. zHowever, triangles can be proved congruent without showing 3 pairs of congruent sides and angles. Congruent Triangles

The Triangle Congruence Postulates &Theorems LA HALL HL FOR RIGHT TRIANGLES ONLY AASASA SAS SSS FOR ALL TRIANGLES

Theorem zIf 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°.

A closer look... zIf 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? 85°30° 85°30°

Example Why aren’t these triangles congruent? What do we call these triangles?

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

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

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

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

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!

HL (Hypotenuse, Leg) zIf both hypotenuses and a pair of legs of two RIGHT triangles are congruent,... A C B F E D then the 2 triangles are CONGRUENT!

HA (Hypotenuse, Angle) zIf both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent,... then the 2 triangles are CONGRUENT! F E D A C B

LA (Leg, Angle) zIf both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent,... then the 2 triangles are CONGRUENT! A C B F E D

LL (Leg, Leg) zIf both pair of legs of two RIGHT triangles are congruent,... then the 2 triangles are CONGRUENT! A C B F E D

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

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

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

Example 4 z Why are the two triangles congruent? z What are the corresponding vertices? A B C D E F SAS  A   D  C   E  B   F

Example 5 zWhy are the two triangles congruent? zWhat are the corresponding vertices? A B C D SSS  A   C  ADB   CDB  ABD   CBD

Example 6 zGiven: B C D A Are the triangles congruent? SSSSSS Why?

Example 7 zGiven: RHSRHS n Are the Triangles Congruent?  QSR   PRS = 90° Q R S P T m  QSR = m  PRS = 90° Why?

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.

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!!!