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Transcript

Proving Triangles Congruent Powerpoint hosted on www.worldofteaching.com Please visit for 100’s more free powerpoints Why do we study triangles so much? • Because they are the only “rigid” shape. • Show pictures of triangles The Idea of a Congruence Two geometric figures with exactly the same size and shape. F B A C E D How much do you need to know. . . . . . about two triangles to prove that they are congruent? Corresponding Parts If all six pairs of corresponding parts of two triangles (sides and angles) are congruent, then the triangles are congruent. 1. AB DE 2. BC EF 3. AC DF 4. A D 5. B E 6. C F ABC DEF Do you need all six ? NO ! SSS SAS ASA AAS Side-Side-Side (SSS) 1. AB DE 2. BC EF 3. AC DF ABC DEF Order matters!! Distance Formula • The Cartesian coordinate system is really two number lines that are perpendicular to each other. • We can find distance between two points via……….. • Mr. Moss’s favorite formula a2 + b2 = c2 Distance Formula • The distance formula is from Pythagorean’s Theorem. • Remember distance on the number line is the difference of the two numbers, so a x2 x1 • And b y2 y1 • So d x2 x1 y2 y1 2 2 To find distance: • You can use the Pythagorean theorem or the distance formula. • You can get the lengths of the legs by subtracting or counting. • A good method is to align the points vertically and then subtract them to get the distance between them • Find the distance between (-3, 7) and (4, 3) 3,7 4 , 3 _____ 7,4 d 7 4 2 2 49 16 65 • Do some examples via Geo Sketch Class Work • Page 240, # 1 – 13 all Warm Up • Are these triangle pairs congruent by SSS? Give congruence statement or reason why not. A F B E G D H C Q N J R M I L K O P Side-Angle-Side (SAS) 1. AB DE 2. A D 3. AC DF ABC DEF included angle Included Angle The angle between two sides G I H Included Angle Name the included angle: E Y S YE and ES E ES and YS S YS and YE Y Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C D NOT CONGRUENT F Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT Why no SSA Postulate? • We would get in trouble if someone spelled it backwards. (bummer!) • Really – SSA allows the possibility of having two different solutions. • Remember: It only takes one counter example to prove a conjecture wrong. • Show problem on Geo Sketch * A Right Δ has one Right . Hypotenuse – Δ Leg - Longest side of a Rt. - Opp. of Right Right Leg – one of the sides that form the right of the Δ. Shorter than the hyp. Th(4-6) Hyp – Leg Thm. (HL) • If the hyp. & a leg of 1 right Δ are to the hyp & a leg of another rt. Δ, then the Δ’s are . Group Class Work • Do problem 18 on page 245 Class Work • Pg 244 # 1 – 17 all Warm Up • Are these triangle pairs congruent? State postulate or theorem and congruence statement or reason why F G not. A J B I E M H C D K L N Q R U V W O P T S Y X Angle-Side-Angle (ASA) 1.A D 2.AB DE 3.B E ABC DEF included side Included Side The side between two angles GI HI GH Included Side Name the included angle: E Y S Y and E YE E and S ES S and Y SY Angle-Angle-Side (AAS) 1. A D 2. B E ABC DEF 3. BC EF Non-included side Angle-Angle-Side (AAS) • The proof of this is based on ASA. • If you know two angles, you really know all three angles. • The difference between ASA and AAS is just the location of the angles and side. • ASA – Side is between the angles • AAS –Side is not between the angles Warning: No SSA Postulate There is no such thing as an SSA postulate! E B F A C D NOT CONGRUENT Warning: No AAA Postulate There is no such thing as an AAA postulate! E B A C D NOT CONGRUENT F Triangle Congruence (5 of them) SSS postulate ASA postulate SAS postulate AAS HL theorem Theorem Right Triangle Congruence When dealing with right triangles, you will sometimes see the following definitions of congruence: HL - Hypotenuse Leg HA – Hypotenuse Angle (AAS) LA - Leg Angle (AAS of ASA) LL – Leg Leg Theorem (SAS) Name That Postulate (when possible) SAS SSA ASA SSS Name That Postulate (when possible) AAA SAS ASA SSA Name That Postulate (when possible) Reflexive Property SAS Vertical Angles SAS Vertical Angles SAS Reflexive Property SSA Name That Postulate (when possible) Name That Postulate (when possible) Let’s Practice Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B D For SAS: AC FE For AAS: A F Review Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS: Class Work • Pg 251, # 1 – 17 all CPCTC • Proving shapes are congruent proves ALL corresponding dimensions are congruent. • For Triangles: Corresponding Parts of Congruent Triangles are Congruent • This includes, but not limited to, corresponding sides, angles, altitudes, medians, centroids, incenters, circumcenters, etc. They are ALL CONGRUENT! Warm Up Given that triangles ABC and DEF are congruent, and G and H are the centroids, find AH. AG = 7x + 8 JH = 5x - 8 Find DJ B I G A E C J H D F