Lesson 4-5: Other Methods of Proving Triangles Congruent (page 140) Essential Questions Can you construct a proof using congruent triangles?

Corresponding Parts of Congruent Triangles are Congruent Please do not forget … CPCTC means … Corresponding Parts of Congruent Triangles are Congruent Ways to Prove Triangles Congruent: SSS Postulate SAS Postulate ASA Postulate

Theorem 4-3 AAS Theorem If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. A Given: ∠B ≅∠E ∠C ≅∠F Prove: ∆ ABC ≅∆ DEF D C B F E

∠C ≅ ∠F A Given: ∠B ≅ ∠E Prove: ∆ ABC ≅ ∆ DEF D C Proof: B F E Statements Reasons

NOTE: Two-column proofs may be shortened by writing them in paragraph form which emphasize the key steps in the proof. Please note that you will be exposed to each type of proof, but you will be responsible to know how to use two-column proofs. When given a choice, use a two column proof.

right sides hypotenuse RIGHT TRIANGLE: a triangle with one right angle. HYPOTENUSE: in a right triangle, the side opposite the ___________ angle. LEGS (of a right triangle): the other two __________. right sides A hypotenuse leg hypotenuse C B leg

Theorem 4-4 HL Theorem If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent. B E A D C F

Ways to Prove ANY Two Triangles Congruent: SSS Postulate SAS Postulate ASA Postulate AAS Theorem Ways that will NOT Prove Triangles Congruent: SSA - except for a special case - what is that? AAA - only works for car issues … flat tire, dead battery, out of gas, trip directions, etc.

Ways to Prove Two RIGHT Triangles Congruent: (Look at the Classroom Exercises on page 143, #14.) HL Theorem LL Method SAS Postulate can be used to prove the ∆’s congruent. ∴ the LL Method is NOT needed!

Ways to Prove Two RIGHT Triangles Congruent: (Look at the Classroom Exercises on page 143, #14.) HL Theorem LL Method HA Method AAS theorem can be used to prove the ∆’s congruent. ∴ the HA Method is NOT needed!

Ways to Prove Two RIGHT Triangles Congruent: (Look at the Classroom Exercises on page 143, #14.) HL Theorem LL Method HA Method LA Method ASA Post. or AAS Thm can be used to prove the ∆’s congruent. ∴ the LA Method is NOT needed!

Ways to Prove ANY Two Triangles Congruent: SSS Postulate SAS Postulate ASA Postulate AAS Theorem Ways to Prove Two RIGHT Triangles Congruent: HL Theorem LL Method - not needed HA Method - not needed LA Method - not needed

Example # 1. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. none __________

Example # 2. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. HL Thm __________

Also SAS Post and ASA Post. AAS Thm Example # 3. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. Also SAS Post and ASA Post. AAS Thm __________

Look at the non-overlapping ∆’s! Look at the overlapping ∆’s! Example # 4. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. W Z X Y Look at the non-overlapping ∆’s! Look at the overlapping ∆’s! AAS Thm HL Thm

Look at overlapping ∆’s! Example # 5. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. C E D A B Look at overlapping ∆’s! ∠A ≅ ∠C AE = DC AAS Thm __________

Example # 6. State which congruence method(s) can be used to prove the triangles congruent. If no method applies, write none. P T S R Q none __________

Can you construct a proof using congruent triangles? Assignment Written Exercises on pages 143 to 145 RECOMMENDED: 5, 6, 7, 10, 12 REQUIRED: 1, 2, 3, 14, 16, 18 Prepare for Quiz on Lessons 4-4 and 4-5 Can you construct a proof using congruent triangles?