GEOMETRY 4.5 Recall: To Prove Triangles are CONGRUENT:

GEOMETRY 4.5 Recall: To Prove Triangles are CONGRUENT: Prove CORRESPONDING Sides and CORRESPONDING Angles are CONGRUENT

GEOMETRY 4.5 Recall: To Prove Triangles are CONGRUENT: Prove CORRESPONDING Sides and CORRESPONDING Angles are CONGRUENT Prove COORESPONDING SIDES are CONGRUENT (SSS)

GEOMETRY 4.5 Recall: To Prove Triangles are CONGRUENT: Prove CORRESPONDING Sides and CORRESPONDING Angles are CONGRUENT Prove COORESPONDING SIDES are CONGRUENT (SSS) Prove Two CORRESPONDING Angles and the INCLUDED SIDE are CONGRUENT (ASA)

GEOMETRY 4.5 Recall: To Prove Triangles are CONGRUENT: Prove CORRESPONDING Sides and CORRESPONDING Angles are CONGRUENT Prove COORESPONDING SIDES are CONGRUENT (SSS) Prove Two CORRESPONDING Angles and the INCLUDED SIDE are CONGRUENT (ASA) Prove Two CORRESPONDING Sides and the INCLUDED ANGLE are CONGRUENT (SAS)

GEOMETRY 4.5 Recall: To Prove Triangles are CONGRUENT: Prove CORRESPONDING Sides and CORRESPONDING Angles are CONGRUENT Prove COORESPONDING SIDES are CONGRUENT (SSS) Prove Two CORRESPONDING Angles and the INCLUDED SIDE are CONGRUENT (ASA) Prove Two CORRESPONDING Sides and the INCLUDED ANGLE are CONGRUENT (SAS) ????

GEOMETRY 4.5 G N Given: N  D G  I AN = SD S A Prove: ANG  SDI D I  

GEOMETRY 4.5 THEOREM: (AAS) If TWO ANGLES and a NONIncluded SIDE of One Triangle are CONGRUENT to the CORRESPONDING TWO ANGLES and SIDE of a Second Triangle, THEN the TWO TRIANGLES are CONGRUENT.

GEOMETRY 4.5 S T Given: PSU  PTR SU = TR R U Prove: P

GEOMETRY 4.5 Prove: Given:

GEOMETRY 4.5 Given: Prove:

GEOMETRY 4.5 Given: E is Midpoint Prove:

GEOMETRY 4.5 Given: Prove:

GEOMETRY 4.5 GEOMETRY 4.5 Summary of Congruency Proof Tools Postulates & Theorems that PROVE Congruency: Postulates/Theorems that DO NOT Prove Congruency:

A SONG about CONGRUENCE

C - P - CTC is a Theorem in Geometry

C - P - CTC is a Theorem in Geometry C - P - CTC proves Congruence Easily

C - P - CTC is a Theorem in Geometry C - P - CTC proves Congruence Easily Get S-S-S A-S-A S-A-S or S-A-A

C - P - CTC is a Theorem in Geometry C - P - CTC proves Congruence Easily Get S-S-S A-S-A S-A-S or S-A-A C - P - CTC follows immediately

GEOMETRY 4.5

 SIDES Base ’s THEOREM: GEOMETRY 4.6  SIDES Base ’s THEOREM: If TWO SIDES of a Triangle are CONGRUENT, THEN The ANGLES OPPOSITE those SIDES are CONGRUENT.

 SIDES Base ’s THEOREM: GEOMETRY 4.6  SIDES Base ’s THEOREM: Q Given: Prove: P R

Triangle ISO is isosceles with base SO. mS = 5x – 18 mO = 2x + 21 GEOMETRY 4.6 S O Triangle ISO is isosceles with base SO. mS = 5x – 18 mO = 2x + 21 m S = m O = m  I = I

Triangle DEF is isosceles mE = 2x + 40 mF = 3x + 22 m D = m E = GEOMETRY 4.6 D Triangle DEF is isosceles mE = 2x + 40 mF = 3x + 22 m D = m E = m F = E F

Base ’s  SIDES THEOREM: GEOMETRY 4.6 Base ’s  SIDES THEOREM:

Base ’s  SIDES THEOREM: GEOMETRY 4.6 Base ’s  SIDES THEOREM: If TWO ANGLES of a Triangle are CONGRUENT, THEN The SIDES OPPOSITE those ANGLES are CONGRUENT.

Base ’s  SIDES THEOREM: GEOMETRY 4.6 Base ’s  SIDES THEOREM: Q Given: Prove: P R

Equilateral Equiangular Theorem GEOMETRY 4.6 TWO COROLLARIES Equilateral Equiangular Theorem A Triangle is EQUILATERAL if and only if It is EQUIANGULAR

Equilateral Equiangular Theorem GEOMETRY 4.6 TWO COROLLARIES Equilateral Equiangular Theorem A Triangle is EQUILATERAL if and only if It is EQUIANGULAR Equilateral Angles are 60 Degrees Theorem Each angle of an EQUILATERAL Triangle measures 60 degrees

D GEOMETRY 4.6 A Given: E C Prove: B

GEOMETRY 4.6

GEOMETRY 4.6

GEOMETRY 4.6

GEOMETRY 4.6

GEOMETRY 4.6

GEOMETRY 4.6

GEOMETRY 4.6

GEOMETRY 4.6

GEOMETRY 4.6

GEOMETRY 4.6 E D C 4 Prove: 3 2 1 A B

GEOMETRY 4.6 W Y Z X V PROVE:

GEOMETRY 4.6