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4.0 Students prove basic theorems involving congruence and similarity. 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles. Agenda for CS 4.0 & 5.0 Although these standards cover both congruency and similarity, this presentation will focus on the congruency aspect and proving triangles are congruent of CS 4.0 & CS5.0 The concept of Cloning is helpful in learning Geo Standards 4 & 5. Cloning is about making exact copies Geometry uses the concept of congruency (exactly the same size) Both Cloning & Geometry focus on Corresponding parts (organs) and their DNA What we know about CLONES… • Clones have the same…DNA • Clones have the same physical characteristics(parts) • Corresponding Parts of Clones are the same (aka Congruent) Therefore, we proved that these What we know about triangles must be congruent Congruent Triangles… because all their corresponding parts congruent. (aka CLONED TRIANGLES) • Congruent triangles Can youare see Are these two have the same DNA… Like Clones, these the connection (Dimension, Numbers, Angles) triangles triangles have between congruent? • Congruent Triangles corresponding have the same (parts) Clones and parts that are physical characteristics Do these Congruent exactly thehave same s3 triangles Triangles? • Corresponding Parts size, thetherefore same DNA? of Congruent they must be (Explain…) Triangles are Congruent Congruent (aka CPCTC) Triangles. s1 s1 s3 s2 s2 Proving Clones using DNA… Evidence is analyzed to compare specific strands of the DNA to determine… 1) Inclusion 2) Exclusion 3) Partial Inclusion* We will use the partial inclusion to *(new of tracking discussmethod similar triangles down criminals) Proving Congruent Triangles… Euclidean Geometry provides five inclusive theorems and postulates for students to prove triangles are congruent. I will refer to these as our DNA Strands used for matching. DNA Strand: Hypotenuse-Leg Using the marking Only used in testing on the two two Rightwe Triangles triangles, for congruency know… • The hypotenuse Hyp Hyp ZX fromAB both triangles need to be Leg CB Leg YX congruent. • Hence, One pair of legs need to be ΔABC ΔZXY by congruent HL Theorem DNA Strand: Side-Side-Side Side(AB) Side(XY) • To use this DNA Strand,Side(YZ) you need to Side(BC) know the lengths of Side(CA) Side(ZX) all six sides in the two triangles Hence, • Hence, SSS really ΔABC ΔXYZ bySS means SS, SS, Side-Side-Side congruency DNA Strand: Side-Angle-Side Hence, the The key to Theorem is triangles are angle. the included congruent These are the included Angles, The included because of angle is because they are formed by created the sides. by the pairs S-A-S of congruent sides angle side Side YouSide have to follow the physical pattern of the DNA Strand angle side DNA Strand: Angle-Side-Angle • A D • AC DF • C F A—A S—S A—A Follows the physical “bonds” Visualize the bonds DNA Strand: Angle-Angle-Side Use your visual skills to get a mental A Y A—A ANGLE picture what a C Z ofA—A non-included AB YX S—S side means in this Angle Theorem…the side Hence, Non-Included Side is notbetween ΔACB ΔYZX by the angles as it-Side is in ASA. Angle-Angle A-A-S Look at the Physical Bonds of the DNA. Once again, look at the physical bonds required … Show-Me-That-You-Know Are these two triangles congruent? If yes, then explain why… Yes, Side-Angle-Side Theorem SMTYK A D A—A 1. Why are these two triangles B E A—A congruent? AC DF S—S 2. Write an informal proof listing the Hence, “DNA Strand” you used to prove the triangles congruent. ΔABC are ΔDEF by Angle-Angle-Side SMTYK What Theorem “DNA Strand” is displayed? SMTYK If you look at the physical bonds, then you will notice Angle-Side-Side…(A-S-S)no bueno •* * WE CANNOT PROVE THEY ARE CONGRUENT * * SMTYK SMTYK 50 Sum of the interior angles of a triangle All we can always equals 180 prove is that the Although AF and thattriangles is not might be congruent, enough information to prove a match we cannot prove it. To solve some of the triangle congruency questions you might Sometimes things need to “operate” and are pull the conjoined triangles apart… stuck together The operation was rated “R”here so itto Meet little Joe and Jae, we are Dr.was Blackenstein is needed in the removed from the presentation. see if they are congruent…knife please! Operating However, the math is Room rated “PG13”. O J E A Before Whatthe other operation, piece ofJOE information and JAE do shared we a common need side to know {JE}. Therefore, to prove these we know triangles that are Side JE Given: OJEAJE congruent JE becauseby of“DNA the Reflexive Strand” P.O.C. A-A-S? The Operation was a success O E J J A E Hints when dealing with conjoined triangle… • Reflexive Property XYXY, or A A • Vertical Angles are Congruent • Parallel lines create Congruent Angles • Perpendicular lines create Right Angles • Midpoints of segments are bisectors • When in doubt, separate the triangles Where are the conjoined triangles…A, B or C ? If you failed to realize it was B, then please visit the nurse at lunch time. Separating Conjoined Triangles A S A Remember that A-S-A really means (AA)-(SS)-(AA) We are matching pairs of angles and sides More Conjoined Δ’s Why are the Since the arrows that these The indicates hearttriangles angles are lines are congruent? parallel, these congruent because happy face angles are they are Vertical congruent because of The DNA indicates….A-S-A Angles Alternate Interior Angle Theorem (AIA ) Bisectors & Midpoints R I Given: Point A is the midpoint of segment BI and segment RN A Prove: ΔARBΔANI B Sides N RANA (definition of midpoint—bisect & create segments) Hint-1: Midpoints create segments Angles RAB NAI (Vertical Angles) Sides BAIA (def. of midpoint) Hint-2: Vertical Angles Using CPCTC to prove R I Given: Point A is the midpoint of segment BI and segment RN A Prove: B N B N This true because of CPCTC HINT: We first have to show that the two RANA (definition of midpoint—bisect & create segments) conjoined triangles are congruent, then Angles RAB NAI (Vertical Angles) we know that (like CLONES) the Sides Sides BAIA (def. of midpoint) corresponding parts of Congruent triangles must be congruent. Prove: ΔARBΔANI Conjoined Δ’s in the Coordinate Plane CACA because of Reflex. P.O.C. (20-4) = 16 side Yes, there are Can we congruent prove that triangles there are congruent of because triangles SSS in the diagram? (10-5) = 5 (10-5) = 5 (20-4) = 16 side Summary and Recap • • • • • • • • • Hypotenuse-Leg HL Side-Side-Side SSS Side-Angle-Side SAS Angle-Side-Angle ASA Angle-Angle-Side AAS CPCTC (corresponding parts) Conjoined Δ (split) Reflex, Vertical ’s, bisectors // & Lines, Midpoints QUIZ: Find & Explain Δ’s below ΔDEF ___ Quiz Integrity is establishing congruency with your values and behavior… Thank your for showing your integrity by allowing me to present this information to you. Sincerely, Mr. Blackwood (aka Dr. Blackenstein)