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Geometry Guided Notes Proving Triangle Congruence Using SSS and SAS Name: __________________________________ Date: ______________________ Period: ______ Proving Triangle Congruence Using SSS and SAS To prove that two triangles are congruent you need to show that ________________________ ______________________________________________________________________________ (SSS)__________________________________________________- If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. ̅̅̅̅̅ ≅ _____________, If side 𝑴𝑵 side ̅̅̅̅̅ 𝑵𝑷 ≅ _____________, and side ̅̅̅̅̅ 𝑷𝑴 ≅ _____________, then ΔMNP ≅ ΔQRS (SAS) _________________________________________________- If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. If side ̅̅̅̅ 𝑷𝑸 ≅ _____________, side ∡𝑸 ≅ _____________, and side ̅̅̅̅ 𝑸𝑺 ≅ _____________, then ΔPQS ≅ ΔWXY SAS______________________________________Not SAS_________________________ Geometry Guided Notes Proving Triangle Congruence Using SSS and SAS Name: __________________________________ Date: ______________________ Period: ______ Example #1: State the included angle of the following sides of the given triangle. ΔAEB: ̅̅̅̅ 𝐴𝐸 𝑎𝑛𝑑 ̅̅̅̅ 𝐸𝐵 _____________________ ̅̅̅̅ 𝑎𝑛𝑑 ̅̅̅̅ 𝐴𝐵 𝐸𝐵 _____________________ ΔMNO: ̅̅̅̅̅ 𝑀𝑁 𝑎𝑛𝑑 ̅̅̅̅ 𝑂𝑁 _____________________ ̅̅̅̅̅ 𝑎𝑛𝑑 ̅̅̅̅ 𝑀𝑂 𝑂𝑁 _____________________ Example #2: Decide whether enough information is given to prove that the triangles are congruent. ΔSTU, ΔPUT S P ΔABC, ΔEBD A U T D B C E Proving Triangle Congruence 1. Examine the drawing or make a sketch. 2. Mark any congruent sides and/or angles. 3. Determine what congruence postulate could be used to prove that the triangles are congruent (SSS, SAS, AAS, ASA or HL). 4. List the statements: givens, congruent parts, etc. 5. Fill in the justifications. Geometry Guided Notes Proving Triangle Congruence Using SSS and SAS Name: __________________________________ Date: ______________________ Period: ______ Properties of Congruent Triangles 1. Every Triangle is congruent to itself (reflexive). 2. If ΔABC ≅ ΔPQR, then ΔPQR ≅ ΔABC (symmetric). 3. If ΔABC ≅ ΔPQR and ΔPQR ≅ ΔTUV, then ΔABC ≅ ΔTUV (transitive). A Example #3: Given: ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐷𝐶 ; ̅̅̅̅ 𝐴𝐶 ≅ ̅̅̅̅ 𝐷𝐵 B C Prove: ∆ABC ≅ ∆DCB D 1. What congruence postulate could we use to prove ∆ABC ≅ ∆DCB? 2. What side(s) must we show congruent? 3. What property states that something (side, angle, etc.) is equal to itself? 4. Now that you have answered these questions, complete a 2-column proof. Statements Justifications Geometry Guided Notes Proving Triangle Congruence Using SSS and SAS Name: __________________________________ Date: ______________________ Period: ______ N Example #4: Given: C is the midpoint of ̅̅̅̅ 𝑁𝐵 ; ̅̅̅̅̅ C is the midpoint of 𝑀𝐴 M A C Prove: ∆NMC ≅ ∆BAC B 1. What congruence postulate could we use to prove ∆NMC ≅ ∆BAC? 2. What does the definition of midpoint tell us? 3. What sides are congruent by the definition of midpoint? 4. Can we show that all sides are congruent? If not can we show that at least one angle is congruent? 5. Now that you have answered these questions, complete a 2-column proof. Statements Justifications Geometry Guided Notes Proving Triangle Congruence Using SSS and SAS Name: __________________________________ Date: ______________________ Period: ______ Example #5: Y X ̅̅̅̅ ≅ ̅̅̅̅̅ ̅̅̅̅ || ̅̅̅̅̅ Given: 𝑋𝑌 𝑊𝑍 ; 𝑋𝑌 𝑊𝑍 Prove: ∆WXY ≅ ∆YZW W Z 1. What congruence postulate could we use to prove ∆WXY ≅ ∆YZW? 2. If ̅̅̅̅ 𝑋𝑌 || ̅̅̅̅̅ 𝑊𝑍, then ̅̅̅̅̅ 𝑊𝑌 can be considered a ______________________? 3. If you have answered number 2, then you can determine what parts are congruent. What postulates/Theorems can be used to justify that these parts are congruent? 4. Now that you have answered these questions, complete a 2-column proof. Statements Justifications Geometry Guided Notes Proving Triangle Congruence Using SSS and SAS Name: __________________________________ Date: ______________________ Period: ______ Example #6: D Perpendicular lines form 4 right angles. All right angles are congruent. Given: ̅̅̅̅ 𝐷𝐸 ⊥ ̅̅̅̅ 𝐴𝐶 ; ̅̅̅̅ 𝐷𝐸 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 ̅̅̅̅ 𝐴𝐶 A Prove: ΔAED ≅ ΔCED Statements Justifications E C