4 0 Students prove basic theorems involving congruence

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 & CS 5. 0

The concept of Cloning is helpful in learning Geo Standards 4 & 5. v. Cloning is about making exact copies v. Geometry uses the concept of congruency (exactly the same size) v. 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 Whattriangles we know about Congruent must be congruent because all Triangles… their corresponding parts are congruent. (aka CLONED TRIANGLES) • Congruent triangles Can you see thehave Aresame these two the DNA… Like Clones, these connection (Dimension, triangles. Numbers, Angles) triangles have between Clones congruent? • corresponding Congruent Triangles have parts the. Congruent same (parts) physical and that are exactly the characteristics Do these Triangles? same size, triangles have • therefore Corresponding Parts they must the same DNA? (Explain…) Congruent beof. Congruent Triangles are Triangles. Congruent (aka CPCTC) s 1 s 3 s 2

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 method of tracking down discuss similar triangles 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 triangles, testing two on the two Right Triangles for we know… congruency Hyp ZX from • Hyp The AB hypotenuse both. CB triangles need to Leg YX be congruent. • Hence, One pair of legs need to be congruent ΔABC ΔZXY by HL Theorem

DNA Strand: Side-Side(AB) • To use Side(XY) this DNA Strand, Side(YZ) you need to Side(BC) know the lengths of all Side(CA) Side(ZX) six sides in the two triangles Hence, • Hence, SSS really ΔABC ΔXYZ by means S S, Side-Side S S congruency

DNA Strand: Side-Angle-Side Hence, The key the to Theorem is triangles are angle. the included congruent These are the included Angles, because of angle is The included because they are formed by the created sides. by the pairs of S-A-S congruent sides angle side You. Side have to follow the. Side 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-Side Use your visual skills to get a Y mental picture of A A—A ANGLE what a non-included C Z A—A side means in this AB YX S—S Theorem…the side is Angle not between the Hence, Non-Included Side angles as it is by in ASA. ΔACB ΔYZX Angle-Angle -Side Look at the Physical A-A-S 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 “DNAHence, Strand” you used to prove the triangles are congruent. ΔABC ΔDEF by Angle-Side

SMTYK What Theorem “DNA Strand” is displayed?

SMTYK If you look at the physical bonds, then you will notice Angle-Side…(A-S-S)no bueno • * * WE CANNOT PROVE THEY ARE CONGRUENT * *

SMTYK

SMTYK 50 Sum of the interior angles of a triangle always equals All we can prove Although is that 180 the A F and that triangles is not might be congruent, we enough information to prove cannot prove it. a match

To solve some of the triangle congruency questions you might Sometimes things arethe need to “operate” and pull conjoined triangles apart… stuck together

The operation waswe rated “R” so it if Meet little Joe and Jae, are here to see Dr. was Blackenstein is needed in the from theplease! presentation. they are removed congruent…knife Operating However, the math is. Room rated “PG 13”. O J E A

Before What theother operation, piece JOE of information and JAE do shared we need a common to side {JE}. know Therefore, to prove these we know triangles that are Sidecongruent JE because by Given: OJE AJE of“DNA the Reflexive Strand” A-A-S? P. O. C. The Operation was a success O E J J A E

Hints when dealing with conjoined triangle… • • • Reflexive Property XY XY, 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 (A A)-(S S)-(A A) We are matching pairs of angles and sides

More Conjoined Δ’s Why are the Since the arrows that these The indicates hearttriangles are lines are parallel, these happy congruent because congruent? 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: ΔARB ΔANI B Sides N RA NA (definition of midpoint—bisect & create segments) Hint-1: Midpoints create. Angles) segments Angles RAB NAI (Vertical Sides BA IA (def. of Angles midpoint) Hint-2: Vertical

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 RA NA (definition of midpoint—bisect & create segments) conjoined triangles are congruent, then Angles RAB NAI (Vertical Angles) we know that (like CLONES) the Sides BA IA (def. of midpoint) corresponding parts of Congruent triangles must be congruent. Prove: ΔARB ΔANI

Conjoined Δ’s in the Coordinate Plane CA CA because of Reflex. P. O. C. (20 -4) = 16 side Yes, there are Can we prove congruent that there are triangles congruent triangles becauseinof the diagram? SSS (10 -5) = 5 (20 -4) = 16 side

Summary and Recap • • • Hypotenuse-Leg HL Side-Side SSS Side-Angle-Side SAS Angle-Side-Angle ASA 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)