Similrity nd ongruence urriculum Redy MMG: 201, 220, 221, 243, 244 www.mthletics.com
SIMILRITY N ONGRUN If two shpes re congruent, it mens thy re equl in every wy ll their corresponding sides nd ngles re equl. Similr figures hve the sme shpe, ut not necessrily the sme size. In this ook, it is shown how similr nd congruent shpes cn e useful in solving prolems. Try to nswer these questions now, efore working through the chpter. I used to think: The symol for congruent is /. Wht do you think it mens to sy T / TF? If squre with side length 4cm hs een enlrged y scle fctor of 2, then wht is the side length of the lrge squre? If two tringles re the sme except for one ngle, re they congruent? nswer these questions, fter working through the chpter. ut now I think: The symol for congruent is /. Wht do you think it mens to sy T / TF? If squre with side length 4cm hs een enlrged y scle fctor of 2, then wht is the side length of the lrge squre? If two tringles re the sme except for one ngle, re they congruent? Wht do I know now tht I didn t know efore? J 1 SRIS TOPI
sics ongruent Tringles (/) ongruent tringles re shpes tht re exctly the sme in every wy (side lengths nd interior ngles re ll equl). If even one side or one ngle re not equl, then the tringles re not congruent. ongruent Tringles These tringles re not congruent 60c 7.4 11.4 80c 40c Q 40c R 80c 11.4 7.4 60c P 60c 20 24.5 50c 22.6 70c F N 48c M 22.6 P ngles: + + P + + Q + + R Sides: PQ QR RP ngle is different ll sides nd ngles re equl therefore tringles re congruent. No ngle in T F is equl to + N. ` These tringles re NOT congruent. From ove, nd PQR re congruent. Using the proper nottion, this is written s / PQR. It is importnt to mke sure the ngles mtch when using the / symol. Here is n exmple: Show tht these tringles re congruent 75c 14 M 75c 14 65c 15 40c N 65c 15 40c P In nd MNP: + + M 75c + + N 65c ngles MN NP sides + + P 40c MP ` / MNP Notice the order of the letters when using /. The equl ngles re written in the sme order. qul ngles written in the sme order (correct): qul ngles not written in the sme order (incorrect): T / TMNP T / TNPM T / TPMN T / TPMN T / TPNM T / TMNP 2 J SRIS TOPI
sics Similr Figures ( ) Similr figures hve the sme shpe, ut not necessrily the sme size. These shpes re similr. cm 5 cm 9 cm cm J 20 cm K 18 cm 15 cm M 30 cm L Similr figures hve two importnt properties: Their corresponding ngles re equl. Their corresponding sides re in the sme rtio. In the ove similr shpes, the rtio of the corresponding sides is 2 since the sides in the igger shpe re doule the length in the smller shpe. These shpes re similr 3 cm P 8 cm 45c Q 24 cm S R 18 cm 135c Find the length of. Find the size of +. PQRS nd re similr. PQRS nd re similr. ` ` PS PQ 24 3 8 ` + + Q ` + 45c ` 9cm c Find the size of + R. d Find the size of RS. PQRS nd re similr. ` + R + ` + R 135c PQRS nd re similr. ` RS RS PQ 8 ` 18 24 ` RS 6cm J 3 SRIS TOPI
Questions sics 1. Show these tringles re congruent, nd then use / symol to stte congruency. T F 14 8 13 13 13 V 8 U 14 G 67c 67c 67c F c 60c 25c F 95c 25c d P 23c 12 13 M 5 67c 12 23c 13 K R 67c 5 Q L 4 J SRIS TOPI
Questions sics 2. Find the missing vlues in these similr shpes (ll mesurements in cm): G 6 85c 7 1c 95c 9 70c F S 14 P Q 12 6 130c 1c 9 150c 15 T S 0c P 3 50c Q 18 R R PQ QR PT ST RS + P + + P + S + Q QR + c L d F M J 4 K 40c 8 15 H 5 G N M 12 45c L K LM Given LM 2. Find the length of KN. F + M + J 5 SRIS TOPI
Knowing More Testing for ongruent Tringles ongruent tringles hve ll 3 corresponding sides equl, nd ll 3 corresponding ngles equl tht is 6 properties. However, there re tests for congruent tringles tht don t require showing ll 6 properties. There re four tests: Side Side Side (SSS) If the corresponding sides of two tringles re equl, then the tringles re congruent (SSS). Q Side ngle Side (SS) If 2 sides nd the included ngle re respectively equl, then the tringles re congruent (SS). M P R L N In nd PQR: PQ QR PR In nd LMN: LM + + L LN T / TPQR (SSS) T / TLMN (SS) Side ngle ngle (S) If 2 corresponding sides nd corresponding ngle re equl, then the tringles re congruent (S). K Right ngle, Hypotenuse, Side (RHS) If two right ngled tringles hve the sme hypotenuse, nd corresponding side, then the tringles re congruent (RHS). N L M M O In nd KLM: KL + + K + + M In nd NOM: NM NO + + M 90c T / TKLM (S) T / TNOM (RHS) 6 J SRIS TOPI
Knowing More Here re some exmples: Show tht these tringles re congruent: L I In TIJK 62c + J 180c-93c- 62c (ngle sum of tringle) 25c 93c M K 93c 12 cm 12 cm 25c ` + N + J (oth re 25c ) N J In T LMN nd T IJK : + N + J + M + K 93c MN KJ 12cm ` TLMN / TIKJ (Proved ove) (Given) (Given) (S) In T F nd T GF : F GF (Given) G (Given) G F is common F ` TF / TGF (SSS) Here is n exmple where congruence is used to show something is true. Show tht isects + in the digrm elow In T nd T : (Given) is common + + 90c (Given) ` + / + (RHS) ` + + (ongruent tringles, T / T ) ` isecting + J 7 SRIS TOPI
Knowing More Similr Tringles ( ) There re two wys to show tht tringles re similr: Show tht their corresponding sides re in proportion. Show tht they hve equl ngles (). If two tringles re similr, the symol is used. Show tht these tringles re similr: In : 78c + 180c - 58c - 78c 44c (ngle sum of tringle) In GHI: 58c +G 180c - 58c - 44c 78c (ngle sum of tringle) G 44c H In nd GHI: + +H (oth re 44c ) 58c + +G (oth re 78c ) I + +I (oth re 58c ) ` GIH () Q 12 R 22 In QRS nd TUV: TU RQ 18 12 3 2 18 S U UV RS TV QS 33 22 27 18 3 2 3 2 18 ` ll sides in proportion TU UV TV c m RQ RS QS T 33 ` QRS TUV 27 V 8 J SRIS TOPI
Questions Knowing More 1. xplin wht the following men: / SSS c SS d S e RHS f J 9 SRIS TOPI
Questions Knowing More 2. Prove these tringles re congruent: 5 5 4 F 3 c P S 75c 75c R M 75c N Q J SRIS TOPI
Questions Knowing More 3. In the digrm elow, show tht. 4. Prove tht JKL STU. J T 8 15 L 6 K S 12 U J 11 SRIS TOPI
Using Our Knowledge Scle Fctor in Similr Tringles When tringles re similr, their ngles re equl () nd their corresponding sides re in proportion. The rtio tht their sides re in proportion is clled the Scle Fctor. Scle Fctor either enlrges (scles up) or reduces (scles down). G 2 4 3 F Scle fctor 1 2 4 8 6 Scle fctor 2 H 8 12 16 I In nd F: In nd GHI: F 3 6 1 2 GI 12 2 6 F 4 8 1 2 scle fctor of F to HI 16 2 8 scle fctor of GHI to 2 4 1 2 GH 8 2 4 ` T T F ` T TGHI If the scle fctor is igger thn 1, the tringle is enlrged. If the scle fctor is etween 0 nd 1 (deciml or frction), the tringle is reduced. Show these tringles re similr nd find their scle fctor of PQR to LMN 6cm N 3cm M 7cm L In LMN nd PQR: PQ LM QR MN 21 3 7 9 3 3 18cm P RP NL 18 3 6 ` LMN PQR (orresponding sides re in proportion) R 21cm PQ QR RP 3 LM MN NL 9cm ` The scle fctor of PQR to LMN is 3. Q 12 J SRIS TOPI
Using Our Knowledge Using Similr Tringles If tringles re known to e similr, then the properties of similr tringles cn e used to solve prolems. Find the vlues of x nd y if TUV (ll mesurements in cm) x In nd TUV: To find x: T 5 TV TU ` x 18 30 (Similr tringles, TUV) 18 30 ` x 18 # 30 ` x 6cm V y To find y: U UV TU y ` 30 5 ` y 5# 30 ` y 15cm (Similr tringles, TUV) J 13 SRIS TOPI
Questions Using Our Knowledge 1. Find the scle fctor in these pirs of similr tringles for oth the smller nd lrger tringles: Given RST UVW. R 35 S 50 25 U 7 V T 5 W Given. 4 8 6 3 14 J SRIS TOPI
Questions Using Our Knowledge 2. nswer these questions out the digrm elow: J 12 N 5 8 K M L Show tht JML JNK. Find the length of KL. c Find the length of ML. J 15 SRIS TOPI
Questions Using Our Knowledge 3. nswer these questions out the shpe elow: F 8 6 G 25 J H I Show tht GFH GIJ. Find the length of GI. c Find the length of IJ. d Find the scle fctor of the lrger tringle with respect to the smll tringle. 16 J SRIS TOPI
Thinking More Using ongruence nd Similrity in Proofs ongruence nd similrity re used to prove properties of tringles, qudrilterls nd other shpes. Show tht the digonls of prllelogrm isect ech other rw in digonls ( nd ) in the prllelogrm : O Given: To prove: O O nd O O Proof: In O nd O ` + + nd + + ` TO / TO ` O O nd O O (Given) (lternte ngles; ) (lternte ngles; ) (Given) (S) (ongruent tringles; TO / TO ) ` The digonls of prllelogrm isect ech other. Similrity cn lso e used in proofs. In the digrm elow, prove tht if Given: To prove: Proof: In nd ` T / T ` / T ` (Given) (orresponding sides in proportion) (Similr tringles; T T ) (Given) J 17 SRIS TOPI
Questions Thinking More 1. nswer these questions out PQRS elow given tht PQ RS nd PR QS: P Q R S Prove PRS / SQP. Prove PQRS is prllelogrm (PQ RS nd PS QR). c Prove tht the opposite ngles of prllelogrm re equl. 18 J SRIS TOPI
Questions Thinking More 2. KLM is n iscosceles tringle with KL KM. K M N L KN hs een drwn to isect ML. Show tht KMN / KLN. Show tht +MNK +LNK 90c. c Prove tht +M +L. 3. In F, +F +, if the line G isects +F. Prove F is isosceles. G F J 19 SRIS TOPI
Questions Thinking More 4. Prove the following out the Rhomus STUV elow: S T O V VOS / TOU U SOT / UOV c Show tht the digonls isect ech other. d Show tht the digonls isect t 90c. 20 J SRIS TOPI
nswers sics: Knowing More: 2. PQ 12cm QR 18cm RS 20cm + P 85c + S 95c + Q 1c 1. e RHS mens Right ngle, Hypotenuse, Side. It is one of the four tests tht cn e used to prove two tringles re congruent. If two right ngle tringles hve equl hypotenuse nd n equl corresponding side, then the tringles re congruent. PT 2cm ST 4cm + 0c + P 1c QR 5cm + 50c f symol mens is similr to. It is used to show two tringles hve the sme shpe (corresponding ngles re equl nd corresponding sides re in proportion). c LM cm 2 2 cm 3 + M 45c + 40c 1. Using Our Knowledge: Scle fctor from RST to UVW is 1 5 Scle fctor from UVW to RST is 5 d KN cm Scle fctor from to is 1 2 1 Knowing More: Scle fctor from to is 3 2 1. / symol mens is congruent to. It is used to show two tringles re exctly the sme in every wy (corresponding sides equl nd corresponding ngles equl). 2. c KL 6cm ML 12cm SSS mens Side, Side, Side. It is one of the four tests tht cn e used to prove two tringles re congruent. If the corresponding sides of two tringles re equl, then the tringles re congruent. 3. c GI 15cm IJ 20cm d Scle fctor from GFH to GIJ is 2 2 1 c SS mens Side, ngle, Side. It is one of the four tests tht cn e used to prove two tringles re congruent. If two sides nd the included ngle of two tringles re equl, then the tringles re congruent. d S mens Side, ngle, ngle. It is one of the four tests tht cn e used to prove two tringles re congruent. If corresponding side nd two corresponding ngles of two tringles re equl, then the tringles re congruent. J 21 SRIS TOPI
Notes 22 J SRIS TOPI
Notes J 23 SRIS TOPI
Notes 24 J SRIS TOPI
www.mthletics.com Similrity nd ongruence