What is AAA similarity theorem
Answers
Answer:
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If the three angles (AAA) are congruent between two triangles, that does not mean that the triangles have to be congruent. They are the same shape (and can be called similar), but we don't know anything about their size.
Prove:
Given : Triangles ABC and DEF such that ∠A = ∠D; ∠B = ∠E; ∠C = ∠F
Prove that : Δ ABC ~ ΔDEF
Construction : We mark point P on the line DE and Q on the line DF such that AB = DP and AC = DQ, we join PQ.
There are three cases :
Case ( i ) : AB = DE, thus P coincides with E.
⇒ AB = DE, BC = EF and AC = DF
Consequently, Q coincides with F.
AB BC CA
---- = ------ = ------
DE EF FA
Since the corresponding angles are equal, we conclude that Δ ABC ~ Δ DEF.
Case( ii ) : AB < DE. Then P lies in DE
In triangles ABC and DPQ,
Statements ------ Reasons
1) AB = DP- 1) By construction
2) ∠A = ∠D -2) Given
3) AC = DQ -3) By construction
4) ΔABC ≅ ΔDPQ -4) By SAS postulate
5) ∠B = ∠DPQ 5) CPCTC
6) ∠B = ∠E- 6) Given
7) ∠E = ∠DPQ- 7) By transitive property
( from above)
8) PQ || EF -8) If two corresponding angles are congruent then the lines are parallel
9) DP/DE = DQ/DF -9) By basic proportionality theorem
10) AB/DE = BC/EF- 10) By construction[tex][/tex]
11) AB/DE = AC/DF- 11) By substitution property
12) Δ ABC ~ Δ DEF - 12) By SAS postulate
