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Unit 5 – Triangle Congruence Day Classwork Day Homework Thursday 11/3 Proving SAS through Rigid Motions 1 HW 5.1 Friday 11/4 Using SAS to Prove Triangles Congruent CPCTC 2 HW 5.2 Monday 11/7 Proving ASA and SSS through Rigid Motions 3 HW 5.3 Tuesday 11/8 AAS, HL AAA, SSA do not determine Congruence Unit 5 Quiz 1 Adding and Subtracting Segments/Angles 4 HW 5.4 5 HW 5.5 Thursday 11/10 Overlapping Triangle proofs Unit 5 Quiz 2 6 HW 5.6 Friday 11/11 No School Monday 11/14 Double Triangle Proofs 7 HW 5.7 Tuesday 11/15 Review Unit 5 Quiz 3 8 Review Packet Wednesday 11/16 Review 9 Review Packet Thursday 11/17 Unit 5 Test 10 Wednesday 11/8 1 Day 1 – Proving SAS through Rigid Motions Recall: We have agreed to use the word ‘congruent’ to mean ‘there exists a _________________________ of basic rigid motions of the plane that maps one figure to the other.’ We are going to show that there are criteria that refer to a few parts of the two triangles and a correspondence between them that guarantee congruency (i.e., existence of rigid motion). We start with the Side-Angle-Side (SAS) criteria. In the figure below, 2 pairs of sides are congruent and the included angles are congruent. AB ____________ AC __________ There are 3 rigid motions that will map symbols . ABC to A ___________ A ' B ' C ' . Describe each rigid motion in words and 1. 2. 3. A B B''' C 2 Examples: 1. Case Diagram Transformations Needed A Shared Side B B''' C B" A Shared Vertex B C C" 2. Given: Triangles with a pair of corresponding sides of equal length and a pair of included angles of equal measure. Sketch and label three phases of the sequence of rigid motions that prove the two triangles to be congruent. Transformation Sketch 3 Directions: Justify whether the triangles meet the SAS congruence criteria; explicitly state which pairs of sides or angles are congruent and why. If the triangles do meet the SAS congruence criteria, describe the rigid motion(s) that would map one triangle onto the other. 1. Given: ∠𝐿𝑁𝑀 ≅ ∠𝐿𝑁𝑂, ̅̅̅̅̅ 𝑀𝑁 ≅ ̅̅̅̅ 𝑂𝑁. Do △ 𝐿𝑀𝑁 and △ 𝐿𝑂𝑁 meet the SAS criteria? ̅. 2. Given: ∠𝐻𝐺𝐼 ≅ ∠𝐽𝐼𝐺, ̅̅̅̅̅ 𝐻𝐺 ≅ 𝐽𝐼 Do △ 𝐻𝐺𝐼 and △ 𝐽𝐼𝐺 meet the SAS criteria? 3. Given: ̅̅̅̅ 𝐴𝐵 ∥ ̅̅̅̅ 𝐶𝐷, ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐶𝐷 . Do △ 𝐴𝐵𝐷 and △ 𝐶𝐷𝐵 meet the SAS criteria? 4 4. Given: 𝑚∠𝑅 = 25°, 𝑅𝑇 = 7", 𝑆𝑈 = 5", 𝑆𝑇 = 5". Do △ 𝑅𝑆𝑈 and △ 𝑅𝑆𝑇 meet the SAS criteria? ̅̅ bisect each other. 5. Given: ̅̅̅̅̅ 𝐾𝑀 and ̅̅ 𝐽𝑁 Do △ 𝐽𝐾𝐿 and △ 𝑁𝑀𝐿 meet the SAS criteria? ̅̅̅̅ bisects angle ∠𝐵𝐶𝐷, ̅̅̅̅ 6. Given: 𝐴𝐸 𝐵𝐶 ≅ ̅̅̅̅ 𝐷𝐶 . Do △ 𝐶𝐴𝐵 and △ 𝐶𝐴𝐷 meet the SAS criteria? 5 Day 2 – Using SAS to Prove Triangles Congruent Describe the additional piece(s) of information needed for each pair of triangles to satisfy the SAS triangle congruence criteria to prove the triangles are congruent. 1. Given: ̅̅̅̅ ≅ ̅̅̅̅ 𝐴𝐵 𝐷𝐶 __________________________ __________________________ Prove: 2. Given: △ 𝐴𝐵𝐶 ≅△ 𝐷𝐶𝐵. ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝑅𝑆 AB RS __________________________ __________________________ Prove: △ 𝐴𝐵𝐶 ≅△ 𝑅𝑆𝑇. Example: We already know that the base angles of an isosceles triangles are congruent. We are going to prove this fact in two ways: (1) by using transformations, and (2) by using SAS triangle congruence criteria. Prove Base Angles of an Isosceles are Congruent Using Transformations ̅̅̅̅ ≅ 𝐴𝐶 ̅̅̅̅ . Given: Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵 Prove: ∠𝐵 ≅ ∠𝐶. 6 Prove Base Angles of an Isosceles are Congruent Using SAS ̅̅̅̅ ≅ 𝐴𝐶 ̅̅̅̅ . Given: Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵 Prove: ∠𝐵 ≅ ∠𝐶. Corresponding parts of congruent triangles are congruent. CPCTC 1. Given: EA AD FD AD F E AC DB AE DF Prove: EC FB A B C D 7 2. Given: KM bisects K LKJ LK JK Prove: M is the midpoint of LJ L J M K 3. Given: Prove: KM LJ LM JM KLJ is isosceles L M J 8 4. Given: C is a midpoint of AE BC DC A C B 3 1 D 2 4 E 1 2 BF DF Prove: AF EF F T 5. Given: Prove: TR TS TM TN 1 2 M 1 R N 2 S 9 Day 3 – Proving ASA and SSS through Rigid Motions A second criteria that guarantees congruency (i.e., existence of rigid motion) is Angle-Side-Angle. 1. In the diagram below, describe the rigid motion that maps congruent by SAS? Why or why not? ABC to ABC ''' . Will the triangles be Since the triangles have two pairs of corresponding congruent angles and the included side congruent, the triangles are congruent by Angle-Side-Angle. 2. Describe the additional piece(s) of information needed for each pair of triangles to satisfy the ASA triangle congruence criteria to prove the triangles are congruent. Given: ABC DCB __________________________ __________________________ Prove: △ 𝐴𝐵𝐶 ≅△ 𝐷𝐶𝐵. A Given: AD BC __________________________ __________________________ Prove: △ 𝐴𝐵𝐷 ≅△ 𝐴𝐶𝐷 B D C 10 3. In the diagram below, describe the rigid motion that maps congruent? Why or why not? ABC to AB ' C . Will the triangles be Since the triangles have three pairs of corresponding congruent sides, the triangles are congruent by Side-Side-Side. 4. Describe the additional piece(s) of information needed for each pair of triangles to satisfy the SSS triangle congruence criteria to prove the triangles are congruent. Given: KM and JN bisect each other at L __________________________ Prove: △ 𝐽𝐾𝐿 ≅△ 𝑁𝑀𝐿. Given: R is the midpoint of KL __________________________ __________________________ Prove: △ 𝐽𝐾𝑅 ≅△ 𝐽𝐿𝑅 11 1. Given: Prove: 2. Given: Prove: ̅̅̅̅, ∠𝐻 ≅ ∠𝑃. 𝑀 is the midpoint of 𝐻𝑃 GHM RPM Rectangle 𝐽𝐾𝐿𝑀 with diagonal ̅̅̅̅̅ 𝐾𝑀. JKM LMK 12 3. Given: Prove: Circles with centers 𝐴 and 𝐵 intersect at 𝐶 and 𝐷. ∠𝐶𝐴𝐵 ≅ ∠𝐷𝐴𝐵. 4. Given: Prove: ∠𝑤 (1) (2) ≅ ∠𝑥 and ∠𝑦 ≅ ∠𝑧. △ 𝐴𝐵𝐸 ≅△ 𝐴𝐶𝐸. ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐴𝐶 and ̅̅̅̅ 𝐴𝐷 ⊥ ̅̅̅̅ 𝐵𝐶 . 13 Day 4 –AAS & HL Triangle Congruences 1. Consider a pair of triangles that meet the AAS criteria (see below). If you knew that two angles of one triangle corresponded to and were equal in measure to two angles of the other triangle, what conclusions can you draw about the third angles of each triangle? Given this conclusion, which formerly learned triangle congruence criteria can we use to determine if the pair of triangles are congruent? Therefore, the AAS criterion is actually an extension of the criterion. triangle congruence Practice: Using only the information given, decide whether the triangles are congruent by ASA or AAS. 1. 3. AB BC DC BC 2. BC AD AB DC 4. 14 5. Consider the two right triangles below that have congruent hypotenuses and one pair of congruent legs. Are the triangles congruent? Why? This is our last criteria to prove triangles congruent: Hypotenuse-Leg (HL) **Note that only RIGHT triangles can be proved congruent using HL** Practice: Using only the information given, decide by which method the triangles could be proved congruent. ED BA 1. 2. 3. HG HI 4. E is the midpoint of CD . A and B are right angles AC DF 15 Criteria that do not determine two triangles as congruent: Two sides and a non-included angle (SSA): Observe the diagrams below. Each triangle has a set of adjacent sides of measures 11 and 9, as well as the non-included angle of 23˚. Yet, the triangles are not congruent. Examine the composite made of both triangles. The sides of lengths 9 each have been dashed to show their possible locations. The pattern of SSA cannot guarantee congruence criteria. In other words, two triangles under SSA criteria might be congruent, but they might not be; therefore we cannot categorize SSA as congruence criterion. Practice: Decide if the triangles are congruent. If so, by what method? 1. Given: A X , BA YX , ZX CA 2. Given: CA ZX , BC YZ , A X 16 Three Congruent Angles (AAA): Observe the diagrams below. What is the measure of the missing angles? Notice that the triangles have 3 pairs of congruent angles. Yet, the triangles are not congruent. 45° 45° The pattern of AAA cannot guarantee congruence criteria. In other words, two triangles under AAA criteria might be congruent, but they might not be; therefore we cannot categorize AAA as congruence criterion. B 1. Given: BD AC A C Prove: ABD CBD A X 1 P 2. Given: PQ RS QO RO Prove: XO YO C D Q O R 2 Y S 17 ADDING & SUBTRACTING SEGMENTS & ANGLES If a pair of congruent segments/angles are added to another pair of congruent segments/angles, then the resulting segments/angles are congruent. Similarly, if a pair of congruent segments/angles are subtracted from a pair of congruent segments/angles, then the resulting segments/angles are congruent. Statements Reasons___________ 1. AB CD 1. Given 2. BC BC 2. Reflexive Property 3. AB BC CD BC 3. Addition Property 4. AC AB BC BD CD BC 4. A whole =’s the sum of its parts 5. AC BD 5. Substitution Statements Reasons___________ 1. AC BD 1. Given 2. BC BC 2. Reflexive Property 3. AC AB BC BD CD BC 3. A whole =’s the sum of its parts 4. AB BC CD BC 4. Substitution 5. AC BD 5. Subtraction Property Examples 1. 18 2. 3. 4. 19 5. 6. 7. 20 8. Given: PS UR PQ UT QR TS Prove: Q T 9. Given: 1 4 2 3 Prove: 5 6 21 Day 6 & 7 – Overlapping and Double Triangle Proofs 1. Given: Prove: ̅̅̅̅ ⊥ 𝐵𝐶 ̅̅̅̅ , 𝐵𝐶 ̅̅̅̅ ⊥ 𝐷𝐶 ̅̅̅̅ 𝐴𝐵 ̅̅̅̅ 𝐷𝐵 bisects ∠𝐴𝐵𝐶, ̅̅̅̅ 𝐴𝐶 bisects ∠𝐷𝐶𝐵 ̅̅̅̅ ̅̅̅̅ ≅ 𝐸𝐶 𝐸𝐵 △ 𝐵𝐸𝐴 ≅ ∆𝐶𝐸𝐷 2. Given: ̅̅̅̅ 𝐵𝐹 ⊥ ̅̅̅̅ 𝐴𝐶 , ̅̅̅̅ 𝐶𝐸 ⊥ ̅̅̅̅ 𝐴𝐵 ̅̅̅̅ ≅ ̅̅̅̅ 𝐴𝐸 𝐴𝐹 Prove: △ 𝐴𝐶𝐸 ≅ ∆𝐴𝐵𝐹 22 ̅̅̅ ≅ 𝑌𝐾 ̅̅̅̅ , 𝑃𝑋 ̅̅̅̅ ≅ 𝑃𝑌 ̅̅̅̅ , ∠𝑍𝑋𝐽 ≅ ∠𝑍𝑌𝐾 3. Given: 𝑋𝐽 Prove: ̅̅̅ 𝐽𝑌 ≅ ̅̅̅̅ 𝐾𝑋 ̅̅̅ ≅ 𝐽𝐿 ̅ , 𝐽𝐾 ̅̅̅ ∥ 𝑋𝑌 ̅̅̅̅ 4. Given: 𝐽𝐾 ̅̅̅̅ ≅ ̅̅̅̅ Prove: 𝑋𝑌 𝑋𝐿 23 5. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4 Prove: ̅̅̅̅ 𝐴𝐶 ≅ ̅̅̅̅ 𝐵𝐷 6. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4, ̅̅̅̅ 𝐴𝐵 ≅ ̅̅̅̅ 𝐴𝐶 Prove: (a) △ 𝐴𝐵𝐷 ≅△ 𝐴𝐶𝐷 (b) ∠5 ≅ ∠6 24 ̅̅̅̅ ≅ 𝐴𝐶 ̅̅̅̅ , 7. Given: 𝐴𝐵 ̅̅̅̅ 𝑅𝐵 ≅ ̅̅̅̅ 𝑅𝐶 , ̅̅̅̅ ≅ 𝑆𝐶 ̅̅̅̅ Prove: 𝑆𝐵 ̅ , ̅̅̅ 8. Given: ̅̅̅ 𝐽𝐾 ≅ 𝐽𝐿 𝐽𝑋 ≅ ̅̅̅ 𝐽𝑌 ̅̅̅̅ ≅ 𝐿𝑌 ̅̅̅̅ Prove: 𝐾𝑋 25 ̅̅̅̅ ⊥ ̅̅̅̅ ̅̅̅̅ ⊥ 𝐵𝑅 ̅̅̅̅, 9. Given: 𝐴𝐷 𝐷𝑅, 𝐴𝐵 ̅̅̅̅ 𝐴𝐷 ≅ ̅̅̅̅ 𝐴𝐵 Prove: ∠𝐷𝐶𝑅 ≅ ∠𝐵𝐶𝑅 26