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Name_________________________________ Geometry Period _____ 5-3 Notes Date______ 5-3: Triangle Congruence and Proof Pieces Learning Goals: What does it mean for 2 triangles to be congruent? What can we infer based on knowing that 2 triangles are congruent? What important aspects are included in two-column proofs? Warm-Up 1. From our homework last night… What are some things you can conclude from the diagrams given? *Can we conclude that these two triangles are congruent? Why or why not? *So, if given ΔABC ≅ ΔDEF, we can identify the pairs of congruent corresponding parts. Corresponding Angles Corresponding Sides <A ̅̅̅̅ 𝐴𝐵 <B ̅̅̅̅ 𝐵𝐶 <C ̅̅̅̅ 𝐴𝐶 ΔABC is congruent to ΔDEF because ___________________________ Therefore, __________________________. ___________________________ . Congruent Triangles are triangles with ______________ corresponding angles and _______________ _____________ sides. Summary of Corresponding Parts Example 1) If given the following picture, what congruence statement (conclusion) can be made? Justify your answer. Example 2) Mark the following diagram based on the given information. Given: FE//AB,̅̅̅̅̅ 𝐹𝐸 ≅ ̅̅̅̅ 𝐴𝐵 , ̅̅̅̅̅ 𝐹𝐶 ≅ ̅̅̅̅̅ 𝐶𝐵 , ̅̅̅̅̅ 𝐸𝐶 ≅ ̅̅̅̅ 𝐴𝐶 Prove: ΔABC ≅ ΔEFC Statements 1. FE//AB,̅̅̅̅̅ 𝐹𝐸 ≅ ̅̅̅̅ 𝐴𝐵 , ̅̅̅̅̅ 𝐹𝐶 ≅ ̅̅̅̅̅ 𝐶𝐵 , ̅̅̅̅̅ ̅̅̅̅ 𝐸𝐶 ≅ 𝐴𝐶 Reasons 1. Let’s work in reverse: Example 1) Given: ΔABC ≅ ΔXYZ ̅̅̅̅? How might we prove that ̅̅̅̅ AC≅ XZ (Hint: what property to do we need to show congruence?) PRACTICE! 2. Identify all pairs of congruent corresponding parts in the triangles below. Then, write a congruence statement for the triangles. Corresponding angles: Corresponding sides: What do you know about these triangles. Explain. 3. Given: ΔABC ≅ ΔXYZ Prove <ACB ≅ <XZY. Statements Reasons 4) In exercises 5-8, ΔXYZ ≅ ΔMNL. Complete each statement and explain how you got each answer! a. m<Y = _____________________________ ___________________________________ b. m<M = _____________________________ ___________________________________ c. m<Z = _____________________________ ___________________________________ d. XY = _______________________________________________________________ 5) CD bisects AB at P. Prove AP = PB. 6) Given: E is the midpoint of AB. Prove: ̅̅̅̅ 𝐴𝐸 = ̅̅̅̅ 𝐸𝐵 7) Find m<1: e. b. 8) Consider the following marking. What was the given? 9) Given: Isosceles triangle ∆𝐴𝐵𝐶 with altitude AB. Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷. Statement Reason 1. 1. Given 2. AB ≅ BC 2. 3. BD ≅ BD 3. 4. BD is a perpendicular bisector and an angle bisector. 4. Altitudes coincide with perpendicular bisectors and angle bisectors in _____________ triangles. 5. AD ≅ DC 5. 6. BD ⊥ AC 6. 7. ∠BDA and ∠ BDC are right angles 7. 8. 8. All right angles are congruent. 9. ∠𝐴𝐵𝐷 ≅ ∠𝐶𝐵𝐷 9. 10. ∠A ≅ ∠C 10. 11. ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷. 12. All corresponding angles are congruent and all corresponding sides are congruent. Name_________________________________ Geometry Period _____ 5-3 HW Date______ 1. List all pairs of congruent corresponding parts. Then, write a congruence statement for the triangles. 2. Find x and y in the following traingles. Explain how you got your answers! 3. Given: CD bisects AB. Prove: AE = EB 4. Given: ̅̅̅̅ 𝑨𝑫 is a perpendicular bisector and ̅̅̅̅ 𝑨𝑫 is an angle bisector. ΔABC is isosceles. Prove that triangle ADC is congruent to triangle ADB.