Theorem tgaascgra

Description: The comparison triangle of the AAS route: with H placed on ray E F at distance B C from E , the triangle D E H is congruent to A B C ( cgrahl2 then tgsas ), so its angle at H can be read off ( cgrcgra ) and matched against the given angle at F . The result is the angle congruence the case split of tgaas1h consumes. (Contributed by JAG, 12-Aug-2026)

	Ref	Expression
Hypotheses	tgaascgra.p	$$⊢P={\mathrm{Base}}_G$$
	tgaascgra.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tgaascgra.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tgaascgra.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	tgaascgra.k	$$⊢K={hl}_{𝒢}\left(G\right)$$
	tgaascgra.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgaascgra.a	$$⊢\phi \to A\in P$$
	tgaascgra.b	$$⊢\phi \to B\in P$$
	tgaascgra.c	$$⊢\phi \to C\in P$$
	tgaascgra.d	$$⊢\phi \to D\in P$$
	tgaascgra.e	$$⊢\phi \to E\in P$$
	tgaascgra.f	$$⊢\phi \to F\in P$$
	tgaascgra.h	$$⊢\phi \to H\in P$$
	tgaascgra.ncol	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
	tgaascgra.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
	tgaascgra.2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
	tgaascgra.3	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EFD”⟩$$
	tgaascgra.4	$$⊢\phi \to HK\left(E\right)F$$
	tgaascgra.5	$$⊢\phi \to E\underset{\dot{}}{-}H=B\underset{\dot{}}{-}C$$
Assertion	tgaascgra	$$⊢\phi \to ⟨“DFE”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DHE”⟩$$

Proof

Step	Hyp	Ref	Expression
1		tgaascgra.p	$$⊢P={\mathrm{Base}}_G$$
2		tgaascgra.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tgaascgra.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tgaascgra.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
5		tgaascgra.k	$$⊢K={hl}_{𝒢}\left(G\right)$$
6		tgaascgra.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
7		tgaascgra.a	$$⊢\phi \to A\in P$$
8		tgaascgra.b	$$⊢\phi \to B\in P$$
9		tgaascgra.c	$$⊢\phi \to C\in P$$
10		tgaascgra.d	$$⊢\phi \to D\in P$$
11		tgaascgra.e	$$⊢\phi \to E\in P$$
12		tgaascgra.f	$$⊢\phi \to F\in P$$
13		tgaascgra.h	$$⊢\phi \to H\in P$$
14		tgaascgra.ncol	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
15		tgaascgra.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
16		tgaascgra.2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
17		tgaascgra.3	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EFD”⟩$$
18		tgaascgra.4	$$⊢\phi \to HK\left(E\right)F$$
19		tgaascgra.5	$$⊢\phi \to E\underset{\dot{}}{-}H=B\underset{\dot{}}{-}C$$
20	1 3 6 5 8 9 7 11 12 10 17	cgracom	$$⊢\phi \to ⟨“EFD”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“BCA”⟩$$
21	1 3 4 6 7 8 9 14	tgbcne	$$⊢\phi \to B\ne C$$
22	1 3 4 6 7 8 9 14	tgacne	$$⊢\phi \to A\ne C$$
23	22	necomd	$$⊢\phi \to C\ne A$$
24		eqid	$$⊢{\sim }_{𝒢}\left(G\right)={\sim }_{𝒢}\left(G\right)$$
25	19	eqcomd	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}H$$
26	1 3 5 6 7 8 9 10 11 12 16 13 18	cgrahl2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEH”⟩$$
27	1 2 3 6 7 8 9 10 11 13 15 26 25	tgsas	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}\left(G\right)⟨“DEH”⟩$$
28	1 2 3 24 6 7 8 9 10 11 13 27	cgr3simp3	$$⊢\phi \to C\underset{\dot{}}{-}A=H\underset{\dot{}}{-}D$$
29	1 2 24 6 8 9 7 11 13 10 25 28 15	trgcgr	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}\left(G\right)⟨“EHD”⟩$$
30	1 3 6 5 8 9 7 11 13 10 21 23 29	cgrcgra	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EHD”⟩$$
31	1 3 6 5 11 12 10 8 9 7 20 11 13 10 30	cgratr	$$⊢\phi \to ⟨“EFD”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EHD”⟩$$
32	1 3 2 6 11 12 10 11 13 10 31	cgraswaplr	$$⊢\phi \to ⟨“DFE”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DHE”⟩$$