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Section 4.2 Triangle Congruence Two triangles are congruent if and only if the corresponding sides and corresponding angles are congruent. E B A C F D ∆ABC ≅ ∆DEF If and only if AC ≅ DF AB ≅ DE BC ≅ EF ∠A ≅ ∠D ∠B ≅ ∠E ∠C ≅ ∠F Theorem 4.1 (SAS Congruence Postulate) If two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent. E B A C D F Theorem 4.2 LL Congruence Theorem If two legs of one right triangle are congruent respectively to two legs of another right triangle, then the two triangles are congruent. B A C E F D Theorem 4.3 HL Congruence Theorem If the hypotenuse and leg of one right triangle are congruent respectively to the hypotenuse and leg of another right triangle, then the two triangles are congruent. B A C E D F Postulate 4.2 ASA Congruence Postulate If two angles and the included side of one triangle are congruent respectively to two angles and the included side of another triangle, then the two triangles are congruent. E B C A F D Postulate 4.3 SSS Congruence If three sides of one triangle are congruent respectively to three sides of another triangle, then the two triangles are congruent. E B A C D F Section 4.4 The Converse of the Pythagorean Theorem If the sum of the squares of the lengths of two sides of a triangle equals the square of the third side, then the triangle is a right triangle. A c b B a C CSCTC Corresponding parts of congruent triangles are congruent. Example 1 N M C D E Let CD ≅ CM and CD ≅ CE . Prove that ∆CMN ≅ ∆CDE and MN ≅ DE Statement Reason 1) CD ≅ CM and CD ≅ CE . Given 2) ∠MCN ≅ ∠DCE If two lines intersect, then the vertical angles formed are congruent 3) ∆MCN ≅ ∆DCE SAS 4) MN ≅ DE CPCTC Example 2 A N C Y D In ANDY, line segments AD and YN divide each in half at C. Prove that ∠DAN ≅ ∠DAY Statement Reason 1) Line segments AD and AD divide each in half at C Given 2) CA ≅ CD and CY ≅ CN A midpoint divide a segment into two congruent segments 3) ∠ACY ≅ ∠NCD and ∠ACN ≅ ∠DCY If two lines intersect, then the vertical angles formed are congruent. 4) ∆ACY ≅ ∆NCD and ∆ACN ≅ ∆DCY SAS 5) AN ≅ BY and AY ≅ DN CPCTC 6) AD ≅ AD Reflective Property of Congruence 7) ∆DYA ≅ ∆AND SSS 8) ∠DAN ≅ ∠DAY CPCTC Example 3 A B D C Given AB ≅ AC and AD is an altitude of ∆ABC , prove that BD ≅ DC A B D C Statement Reason 1) AB ≅ AC and AD is an altitude of ∆ABC 1) Given 2) ∠BDA and ∠CDA are right angles 2) Definition of an altitude of a triangle 3) ∠BDA ≅ ∠CDA 3) All right angles are congruent 4) ∠BDA and ∠CDA are right triangles 4) Definition of a right triangle 5) AD ≅ AD 5) Reflective Property of Congruence 6) ∆BDA ≅ ∆CDA 6) HL 7) BD ≅ DC 7) CPCTC