Difference between revisions of "AA similarity"

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In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.
 
In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.
  
Proof:
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==Proof==
 
Let ABC and DEF be two triangles such that <math>\angle A = \angle D</math> and <math>\angle B = \angle E</math>.
 
Let ABC and DEF be two triangles such that <math>\angle A = \angle D</math> and <math>\angle B = \angle E</math>.
 
<math>\angle A + \angle B + \angle C = 180</math> and
 
<math>\angle A + \angle B + \angle C = 180</math> and
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\angle D + \angle E + \angle C = \angle D + \angle E + \angle F</math>, since we know that <math>\angle A = \angle D</math> and <math>\angle B = \angle E</math>, from before.
 
\angle D + \angle E + \angle C = \angle D + \angle E + \angle F</math>, since we know that <math>\angle A = \angle D</math> and <math>\angle B = \angle E</math>, from before.
 
Therefore, by subtracting <math>\angle D + \angle E</math> by both equations, we get <math>\angle C = \angle F</math>.
 
Therefore, by subtracting <math>\angle D + \angle E</math> by both equations, we get <math>\angle C = \angle F</math>.
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==See also==
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* [[Similarity (geometry)]]
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* [[SAS similarity]]
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* [[SSS similarity]]

Revision as of 01:52, 19 December 2020

Theorem: In two triangles, if two pairs of corresponding angles are congruent, then the triangles are similar.

Proof

Let ABC and DEF be two triangles such that $\angle A = \angle D$ and $\angle B = \angle E$. $\angle A + \angle B + \angle C = 180$ and $\angle D + \angle E + \angle F = 180$ Thus, we can write the equation: $\angle A  + \angle B + \angle C=\angle D + \angle E + \angle F \Longrightarrow \angle D + \angle E + \angle C = \angle D + \angle E + \angle F$, since we know that $\angle A = \angle D$ and $\angle B = \angle E$, from before. Therefore, by subtracting $\angle D + \angle E$ by both equations, we get $\angle C = \angle F$.

See also