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Mirrors > Home > MPE Home > Th. List > tgasa | Structured version Visualization version GIF version |
Description: Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.) |
Ref | Expression |
---|---|
tgsas.p | β’ π = (BaseβπΊ) |
tgsas.m | β’ β = (distβπΊ) |
tgsas.i | β’ πΌ = (ItvβπΊ) |
tgsas.g | β’ (π β πΊ β TarskiG) |
tgsas.a | β’ (π β π΄ β π) |
tgsas.b | β’ (π β π΅ β π) |
tgsas.c | β’ (π β πΆ β π) |
tgsas.d | β’ (π β π· β π) |
tgsas.e | β’ (π β πΈ β π) |
tgsas.f | β’ (π β πΉ β π) |
tgasa.l | β’ πΏ = (LineGβπΊ) |
tgasa.1 | β’ (π β Β¬ (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) |
tgasa.2 | β’ (π β (π΄ β π΅) = (π· β πΈ)) |
tgasa.3 | β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) |
tgasa.4 | β’ (π β β¨βπΆπ΄π΅ββ©(cgrAβπΊ)β¨βπΉπ·πΈββ©) |
Ref | Expression |
---|---|
tgasa | β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπΉββ©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgsas.p | . 2 β’ π = (BaseβπΊ) | |
2 | tgsas.m | . 2 β’ β = (distβπΊ) | |
3 | tgsas.i | . 2 β’ πΌ = (ItvβπΊ) | |
4 | tgsas.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | tgsas.a | . 2 β’ (π β π΄ β π) | |
6 | tgsas.b | . 2 β’ (π β π΅ β π) | |
7 | tgsas.c | . 2 β’ (π β πΆ β π) | |
8 | tgsas.d | . 2 β’ (π β π· β π) | |
9 | tgsas.e | . 2 β’ (π β πΈ β π) | |
10 | tgsas.f | . 2 β’ (π β πΉ β π) | |
11 | tgasa.2 | . 2 β’ (π β (π΄ β π΅) = (π· β πΈ)) | |
12 | tgasa.3 | . 2 β’ (π β β¨βπ΄π΅πΆββ©(cgrAβπΊ)β¨βπ·πΈπΉββ©) | |
13 | tgasa.l | . . 3 β’ πΏ = (LineGβπΊ) | |
14 | tgasa.1 | . . 3 β’ (π β Β¬ (πΆ β (π΄πΏπ΅) β¨ π΄ = π΅)) | |
15 | tgasa.4 | . . 3 β’ (π β β¨βπΆπ΄π΅ββ©(cgrAβπΊ)β¨βπΉπ·πΈββ©) | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 11, 12, 15 | tgasa1 28661 | . 2 β’ (π β (π΅ β πΆ) = (πΈ β πΉ)) |
17 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16 | tgsas 28658 | 1 β’ (π β β¨βπ΄π΅πΆββ©(cgrGβπΊ)β¨βπ·πΈπΉββ©) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β¨ wo 846 = wceq 1534 β wcel 2099 class class class wbr 5148 βcfv 6548 (class class class)co 7420 β¨βcs3 14825 Basecbs 17179 distcds 17241 TarskiGcstrkg 28230 Itvcitv 28236 LineGclng 28237 cgrGccgrg 28313 cgrAccgra 28610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-oadd 8490 df-er 8724 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-s2 14831 df-s3 14832 df-trkgc 28251 df-trkgb 28252 df-trkgcb 28253 df-trkgld 28255 df-trkg 28256 df-cgrg 28314 df-leg 28386 df-hlg 28404 df-mir 28456 df-rag 28497 df-perpg 28499 df-hpg 28561 df-mid 28577 df-lmi 28578 df-cgra 28611 |
This theorem is referenced by: isoas 28667 |
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