OVERVIEW Properties of Similarity & Similarity Criteria G.SRT.3

Transcription

1 OVRVIW Prperties f Similarity & Similarity riteria G.SRT.3 G.SRT.3 Use the prperties f similarity transfrmatins t establish the criterin fr tw triangles t be similar. This bjective has been included in gemetry fr many years develping the criteria fr similarity. While it nly specifically mentins, this is the apprpriate place t als prve that SS and SSS establish similarity. The criterin fr similarity shuld be funded in similarity transfrmatins. (1) The student will be able t prve tw triangles t be similar using the minimum requirements f, SS and SSS. (2) The student will be able t use the prperties f similarity transfrmatins t establish the, SS and SSS criterin fr tw triangles t be similar. The main area f fcus here is t demnstrate hw if tw triangles have tw angles knwn there is a sequence f similarity transfrmatins that will map ne triangle nt anther. Once is dne, SS and SSS fllw quite easily. 1 Use transfrmatins t establish the three similarity criterin,, SSS and SS. 2 Link dilatins and dilatin prperties t the similar triangles as ften as yu can. It will help t link the ld cncepts with the new nes. NS g.srt.3 PrpertiesSimilarity Page 1 f 7 4/29/2014

2 NOTS Prperties f Similarity and Similarity riteria G.SRT.3 ONPT 1 Use the prperties f similarity transfrmatins t establish the criterin fr tw triangles t be similar. When studying cngruent triangles in G.O.8 sme minimum requirements were established that wuld guarantee cngruence thrugh a single r sequence f ismetric transfrmatins. It was fund that SSS, SS, S, S and HL (and sme special cases f SS) were enugh infrmatin t always establish cngruence between tw triangles. In a likewise manner, it is necessary t find the minimum requirements in tw triangles t establish similarity. In rder t d this, use a single r sequence f similarity transfrmatins that wuld map ne triangle nt the ther. egin by seeing if knwing tw cngruent crrespnding angles () is a criterin fr similarity. Given: and Prve: T prve tw triangles t be similar t find a sequence f similarity transfrmatins that map n t. irst dilate by the scale factr, k =, t get. because dilatin is a similarity transfrmatin and using the prperties f a dilatin it is knwn that: and and that = k = () = ( = ) & and & s using the transitive prperty and. ' ' ' and = due t the dilatin. Thus by S. Therefre because was mapped nt using nly similarity transfrmatins. is a similarity criterin. gsrt3prpertiesfsimilarity Page 2 f 7 4/28/2014

3 NOTS Prperties f Similarity and Similarity riteria G.SRT.3 ONPT 2 Use the prperties f similarity transfrmatins t establish the SS criterin fr tw triangles t be similar. Nw investigate if knwing tw crrespnding prprtinal sides and the included crrespnding cngruent angle (SS) is enugh fr establishing similarity. Just as it was imprtant in cngruence the angle must be the included angle meaning that it is the ne between the tw sides. Given: and Prve: T prve tw triangles t be similar find a sequence f similarity transfrmatins that map n t. irst dilate by scale factr, k t get. = k = () = = k = () = () = ( = ) ( = ) ' because f the similarity transfrmatin, which means that: ' ' & s using the transitive prperty, and = and = are equal due t the dilatin. Thus by SS. because was mapped nt using nly similarity transfrmatins. Thus SS is a similarity criterin. gsrt3prpertiesfsimilarity Page 3 f 7 4/28/2014

4 NOTS Prperties f Similarity and Similarity riteria G.SRT.3 ONPT 3 Use the prperties f similarity transfrmatins t establish the SSS criterin fr tw triangles t be similar. Given: Prve: T prve tw triangles t be similar find a sequence f similarity transfrmatins that map n t. irst dilate by scale factr, k t get. = k = () = = k = () = () = = = ' = k = () = () = = ' ' Thus by SSS. because was mapped nt using nly similarity transfrmatins. SSS is a similarity criterin. gsrt3prpertiesfsimilarity Page 4 f 7 4/28/2014

5 SSSSMNT Prperties f Similarity & Similarity riteria G.SRT.3 1. Which f the fllwing wuld be the criterin fr establishing similarity in the tw triangles given? ) ) SS ) SSS ) Nt enugh inf r nt similar 2. Which f the fllwing wuld be the criterin fr establishing similarity in the tw triangles given? ) ) SS ) SSS ) Nt enugh inf r nt similar θ θ 3. If tw sides are prprtinal, then the tw triangles must be similar. T r 4. re the fllwing tw plygns LWYS, SOMTIMS, OR NVR similar? a) Tw quilateral Triangles LWYS SOMTIMS NVR b) Tw Rectangles LWYS SOMTIMS NVR c) Tw Issceles Triangles LWYS SOMTIMS NVR 5. re the fllwing pairs f triangles similar? If they are, then name their similarity criteria. (SSS, SS, ) a) Yes / N b) Yes / N c) Yes / N d) Yes / N e) Yes / N f) Yes / N NS g.srt.3 PrpertiesSimilarity Page 5 f 7 4/29/2014

6 SSSSMNT Prperties f Similarity and Similarity riteria G.SRT.3 6. Prve that if tw triangles have tw cngruent crrespnding angles, then they must be similar. Given: K and L Prve: HKL O H X O K X L gsrt3prpertiesfsimilarity Page 6 f 7 4/28/2014

7 SSSSMNT Prperties f Similarity and Similarity riteria G.SRT.3 nswer: 1) 2) 3) 4) a) lways b) Smetimes c) Smetimes 5) a) Yes, SS b) N c) Yes, SS d) Yes, SS r e) Yes, SS f) Yes, 6) Given: K and L Prve: HKL T prve tw triangles t be similar first find a sequence f similarity transfrmatins that map n t HKL. irst dilate by scale factr, k = KL t get. because dilatin is a similarity transfrmatin and using the prperties f a dilatin: and and = k = KL () = KL ( = KL) & K and & L s using the transitive prperty K and L and = KL due t the dilatin. Thus HKL by S. nd HKL because was mapped nt HKL using nly similarity transfrmatins. is a similarity criterin. gsrt3prpertiesfsimilarity Page 7 f 7 4/28/2014