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Triangle Congruence Packet #2 Name ____________________________ Teacher _____________ 1 Table of Contents Day 1 : SWBAT: Prove Triangles Congruent using SSS and SAS Theorems Pages 3 - 8 HW: Pages 9 - 11 Day 2: SWBAT: Prove Triangles Congruent using ASA and AAS Theorems Pages 12 - 16 HW: pages 17 - 19 Day 3: SWBAT: Prove Triangles Congruent using HL Theorem Pages 20-26 HW: pages 27 - 28 Day 4: SWBAT: Proving Overlapping and Isosceles Triangles Congruent Pages 29 - 34 HW: pages 35 - 37 Day 5: SWBAT: Use CPCTC to Prove Segment Relationships of Triangles Pages 38 - 43 HW: pages 44 - 46 Day 6&7: SWBAT: Review of Proving Triangles Congruent Pages 47- 53 Day 8: Test 2 Day 1 – Proving Triangles Congruent by SSS and SAS Warm - Up Which of the following condition(s) does not prove that two triangles are congruent? 1) SSS ≅ SSS 2) SSA ≅ SSA 3) SAS ≅ SAS 4) ASA ≅ ASA 5) AAA ≅ AAA 3 Model Problem #1 Given: ̅̅̅̅ ̅̅̅̅ D is the midpoint of ̅̅̅̅ Prove: Given Given _____ ______ _____ ______ _________ _________ 2) Given _____ ______ _____ ______ _____ ______ ____________ _________ ____________ _________ 4 3) Given d ___ _________ _____ ______ Given _____ ______ You Try It! 4) 5 ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ Given _____ ______ _____ ______ You Try It! 6 Challenge SUMMARY Example 1: Proving Triangles Congruent 7 SUMMARY Exit Ticket 8 Homework 1. 2. 9 3. 4) Given: ̅̅̅̅ ̅̅̅̅, ⃗⃗⃗⃗⃗ Prove: 10 5) ̅̅̅̅̅ ̅̅̅̅̅ 6. 11 Day 2 – Proving Triangles Congruent by ASA and AAS Warm-Up What additional information would you need to prove these triangles congruent by ASA? _____ ______ Model Problem #1 Prove: (a) YWZ YWX (b) ̅̅̅̅ ̅̅̅̅ Given _____ ______ ___ _________ _____ ______ _____ ______ 12 You Try It! 13 14 Challenge SUMMARY of the ASA Theorem 15 SUMMARY of the AAS Theorem Exit Ticket 16 Day 2 - AAS/ASA Proofs Homework 1. 2. 17 3. 4. Prove: P 18 5. Given: ̅̅̅̅ ̅̅̅̅, T is the midpoint of ̅̅̅̅. Prove: ̅̅̅̅ ̅̅̅̅ \ 5. 6. 19 Day 3 – HL 20 Example 1: 21 You Try It! Example 2: Given: ̅̅̅̅̅ Prove: ̅̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅ 22 Theorem: All radii of a circle are congruent! You Try It! 23 You Try It! Exit Ticket 24 Challenge 25 Summary HL Theorem Proofs Involving Circles 26 Homework 1. 2. Given: ̅̅̅̅ Prove: ̅̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ 27 3. 4. 28 Day 4 – Overlapping and Isosceles Triangles Warm – Up 29 3. Draw the triangles separately here: 4. Draw the triangles separately here: 30 Using Isosceles Triangles in Proofs When we studied triangles in earlier chapters, we learned about the properties of the isosceles triangles. We can use these ideas to help us when constructing proofs. Example 5: 31 You Try It! Given: XYZ is isosceles with base ̅̅̅̅. ̅̅̅̅̅ ̅̅̅̅ Prove: Example 6: Given: ̅̅̅̅ Prove: ̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅ is an Isosceles Triangle 32 Challenge Exit Ticket 33 SUMMARY of Overlapping Triangles \\\\\\\\\\\\\ Summary of Isosceles Triangles 34 Homework 1. 2. 3. 35 4. 5. Given: ∆PMR is isosceles with base ̅̅̅̅̅ Prove: ∆PSM 36 6. Given: ̅̅̅̅ ̅̅̅̅̅ Prove: ̅̅̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅ ̅̅̅̅ 7. 37 Day 5 – CPCTC and Beyond Warm - Up Complete the proof below by filling in the reasons. Statements Reasons 38 Example 1: ADC BDC CPCTC Example 2: Prove: B is the midpoint of ̅̅̅̅ 39 Ex 3: Ex 4: Given: ̅̅̅̅ Prove: ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ 40 Altitudes An altitude of a triangle is a line segment drawn from any vertex of a triangle, perpendicular to the opposite side of the triangle. Ex 5: Given: ̅̅̅̅ Prove: ̅̅̅̅ ̅̅̅̅ ̅̅̅ ̅̅̅ and 41 Proving Perpendicular Segments Theorem: Two angles are congruent and supplementary each angle is a right angle Ex 5: Prove: ̅̅̅̅ ̅̅̅̅̅ Challenge 42 SUMMARY Exit Ticket Which Theorem below would not be used in justifying the congruence below? 43 Homework 1. 2. 44 3. Given: ̅̅̅̅ Prove: ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ is an altitude to ̅̅̅̅ Prove: ̅̅̅̅ bisects 4. Given: 45 ̅̅̅̅ is an altitude to ̅̅̅̅̅ 5. Given: Prove: ̅̅̅̅ bisects ̅̅̅̅̅ 46 Day 6 – REVIEW 47 Concept Summary Addition and Subtraction Use ADDITION when: a) you have a gap. b) you need larger pieces. Use SUBTRACTION when: a) you have overlap. b) you need smaller pieces. CPCTC Stands for: Corresponding Parts of Congruent Triangles are Congruent Use when you are asked to prove SEGMENTS or ANGLES congruent. Proofs involving Circles Theorem: All radii are congruent! Isosceles Triangle Theorem Or Overlapping Triangles Draw triangles separately. Mark up both the together and separate diagrams. Know how to draw conclusions from “Key” Vocabulary Words 1. 2. 3. 4. 5. 6. 7. 8. Midpoint → 2 segments Bisector → 2 segments Angle bisector → 2 angles Perpendicular ( ) → right angles → all right angles are Median → Midpoint → 2 segments Altitude → Perpendicular ( ) → right angles → all right angles are Vertical Angles → angles are → Alternating Interior Angles (A.I.A’s) 48 Geometry Practice Proofs 1. 2. 3. 49 4. 5. 6. 50 ̅̅̅̅ ̅̅̅̅ bisects ̅̅̅̅̅ ̅̅̅̅ 51 ̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅ 52 ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ ̅̅̅̅ 53