Theorem tgaas

Description: Angle-angle-side: two triangles with two congruent angles and a congruent side not included between them are congruent. The side included between the two angles is congruent by tgaas1 , which reduces the theorem to SAS ( tgsas ). (Contributed by JAG, 12-Aug-2026)

	Ref	Expression
Hypotheses	tgaas1.p	$$⊢P={\mathrm{Base}}_G$$
	tgaas1.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
	tgaas1.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	tgaas1.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	tgaas1.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	tgaas1.a	$$⊢\phi \to A\in P$$
	tgaas1.b	$$⊢\phi \to B\in P$$
	tgaas1.c	$$⊢\phi \to C\in P$$
	tgaas1.d	$$⊢\phi \to D\in P$$
	tgaas1.e	$$⊢\phi \to E\in P$$
	tgaas1.f	$$⊢\phi \to F\in P$$
	tgaas1.ncol	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
	tgaas1.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
	tgaas1.2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
	tgaas1.3	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EFD”⟩$$
Assertion	tgaas	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}\left(G\right)⟨“DEF”⟩$$

Proof

Step	Hyp	Ref	Expression
1		tgaas1.p	$$⊢P={\mathrm{Base}}_G$$
2		tgaas1.m	$$⊢\underset{\dot{}}{-}=\mathrm{dist}\left(G\right)$$
3		tgaas1.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
4		tgaas1.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
5		tgaas1.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
6		tgaas1.a	$$⊢\phi \to A\in P$$
7		tgaas1.b	$$⊢\phi \to B\in P$$
8		tgaas1.c	$$⊢\phi \to C\in P$$
9		tgaas1.d	$$⊢\phi \to D\in P$$
10		tgaas1.e	$$⊢\phi \to E\in P$$
11		tgaas1.f	$$⊢\phi \to F\in P$$
12		tgaas1.ncol	$$⊢\phi \to \neg \left(C\in \left(ALB\right)\vee A=B\right)$$
13		tgaas1.1	$$⊢\phi \to A\underset{\dot{}}{-}B=D\underset{\dot{}}{-}E$$
14		tgaas1.2	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“DEF”⟩$$
15		tgaas1.3	$$⊢\phi \to ⟨“BCA”⟩{\sim }_{𝒢}^{\angle }\left(G\right)⟨“EFD”⟩$$
16	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15	tgaas1	$$⊢\phi \to B\underset{\dot{}}{-}C=E\underset{\dot{}}{-}F$$
17	1 2 3 5 6 7 8 9 10 11 13 14 16	tgsas	$$⊢\phi \to ⟨“ABC”⟩{\sim }_{𝒢}\left(G\right)⟨“DEF”⟩$$