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Geometry A Unit 3 Day 4 Notes 5 Shortcuts for Proving Congruence Warm-Up In each of the following, determine the segments or angles that are congruent based on the given information. If two segments or angles are congruent, circle their names. If a second set of segments or angles is congruent, put a box around them. 1. In the diagram below, TS bisects YL. 2. In the diagram below, M is the midpoint of TS. T Y M T M Y S L Which segments are congruent? S Which segments are congruent? TM TM MS YM ML YL ST YM ML YL ST 4. In the diagram below, AB MN. A 3. Y MS L T M S L M Which angles are congruent? YMT LMT LMS YMS N 1 3T 4 2 B Which angles are congruent? 1 2 3 4 6. In the diagram, TR bisects WRY 5. In the diagram, MY | | LT M 3 1 2 4 Y T L 5 78 6 1 2 L 3 R T Which angles are congruent? 1 2 3 4 5 Y W 6 7 8 Which angles are congruent? 1 2 3 I. Proving That Two Triangles are Congruent A. If you KNOW that two triangles share the right number of parts, in the right order, you can be sure they are congruent and use the resulting symmetry as you see fit. 1. ________________ - ___________________ - ________________ Congruence a. Every problem in this chapter is like a proof, but not all will be. We will however start with one. Given: RST is isosceles with base RT and RV = VT Prove: RSV TSV Statement S Reason 1. __________________________ 1. _______________ 2. __________________________ 2. _______________ 3. __________________________ 3. ___________________ R T V 4. ___________________________ 4. __________________ 5. ___________________________ 5. ___________________ Examples: Determine if enough information is given to be sure that the two triangles are congruent. If so, complete the congruence statement. IF not, tell what other pair of parts would need to match. a. b. Given: RST is isosceles with base RT and RV = TV S Given: AB = DE, BC = EG and AC = DG A C D B E Yes: ABC ___ ___ ___ No: We need ______ ________ G R V T Yes: SVR ___ ___ ___ No: We need ______ ________ d. EQL is Equilateral E c. M is the midpoint of NP, NP bisects QR and NR = QP. P T Q L M N Q R Yes: MQP ___ ___ ___ Yes: ETQ ___ ___ ___ No: We need ______ ________ No: We need ______ ________ 2. __________________ - ________________ - _________________ congruence (Notice that the angle is included between the two sides) Examples: Determine if enough information is given to be sure that the two triangles are congruent. If so, complete the congruence statement. IF not, tell what other pair of parts would need to match. b. Given: A P AB PT a. Given AB DE B is the midpoint of DE A A C P B E B D T ABD ___ ___ ___ CBA ___ ___ ___ No: We need ______ ________ No: We need ______ ________ V d. AT | | CU, AT CU, CU bisects AM. c. M is the midpoint of NP NP bisects QR. N A R T C M P U M Q MRN ___ ___ ___ TCA ___ ___ ___ No: We need ______ ________ No: We need ______ ________ In the drawing below, three parts are used to start two “different” triangles. I. A 46 28 B C 46 28 D A. List pairs of equal angles. ____ ____ and ____ ____ B. List a pair of equal sides. _______ = ________ C. Complete the drawings of the two triangles. Are the resulting triangles identical? _____________________ If the answer is yes, then three parts in the order we started with are enough to guarantee two triangles are congruent. _____ _____ _____ Ex. 1: Determine which figure shows triangles that can be proven to be congruent. In the figure where we do not know enough information, give a pair of parts that would convince us the triangles are congruent. a. Given: 1 2 snd AC bisects BCD b. Given: AB BD and EC CD E B B D A 1 2 C C A D YES (circle the reason) SSS SAS ASA or NO we need ________ __________ II. YES (circle the reason) SSS SAS or NO we need ________ __________ The remaining two shortcuts work because they lead directly to one of the three. A. Angle – Angle – Side 1. Find the missing angle in each triangle. 28 46 ASA 46 28 2. If two of the angles in a triangle match two angles in another triangle, the third has to match. So, AAS leads to ____ _____ ____. Therefore, we’ll add AAS to the list. B. Hypotenuse – Leg 1. If we call the sides of a triangle hypotenuse and leg, then the triangle must be a _______________ triangle. 2. Below, two triangles are drawn that have matching hypotenuses and a matching leg. The third side HAS to be 12 in each. (Pythagorean Theorem). 5 13 13 5 3. The sides you found should have matched. Therefore, HL leads to _____ _____ _____. So we should add HL to the list. C. In case you were wondering why we don’t use AAA or SSA, those parts do not necessarily establish one triangle. That is… A picture can be drawn to make it look like triangles are the same because of AAA… But it is possible to draw two triangle of different size even if their angles match. Triangles can be drawn to look like they are the same because of SSA… But it is possible that two different triangles can have two matching sides and a matching angle, so long as the angle between the sides doesn’t match. Therefore the complete list of congruent triangle shortcuts is… 1) __________ 2) _________ 3) __________ 4) __________ 5) _________ HW: Handout