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Difference between revisions of "AA similarity"
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==Proof==
 
==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>.
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<math>\angle A + \angle B + \angle C = 180</math> and
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The sum interior angles of a triangle is equal to 180. Therefore, <math>\angle A + \angle B + \angle C = 180</math>, and
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<math>\angle D + \angle E + \angle F = 180</math>  
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<math>\angle D + \angle E + \angle F = 180</math>.
−
Thus, we can write the equation: <math>\angle A  + \angle B + \angle C=\angle D + \angle E + \angle F \Longrightarrow
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We can write the equation: <math>\angle A  + \angle B + \angle C = 180 = \angle D + \angle E + \angle F \Longrightarrow
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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.
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\angle D + \angle E + \angle C = \angle D + \angle E + \angle F</math>, acknowledging the fact that <math>\angle A = \angle D</math> and <math>\angle B = \angle E</math>.
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Therefore, by subtracting <math>\angle D + \angle E</math> by both equations, we get <math>\angle C = \angle F</math>.
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To conclude, by subtracting <math>\angle D + \angle E</math> by both equations, we get <math>\angle C = \angle F</math>. Since the three angles are congruent, the two triangles are similar.
  
 
==See also==
 
==See also==
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* [[SAS similarity]]
 
* [[SAS similarity]]
 
* [[SSS similarity]]
 
* [[SSS similarity]]
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{{stub}}

Latest revision as of 11:04, 21 September 2024

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 $\angle B = \angle E$. The sum interior angles of a triangle is equal to 180. Therefore, $\angle A + \angle B + \angle C = 180$, and $\angle D + \angle E + \angle F = 180$. We can write the equation: $\angle A + \angle B + \angle C = 180 = \angle D + \angle E + \angle F \Longrightarrow \angle D + \angle E + \angle C = \angle D + \angle E + \angle F$, acknowledging the fact that $\angle A = \angle D$ and $\angle B = \angle E$. To conclude, by subtracting $\angle D + \angle E$ by both equations, we get $\angle C = \angle F$. Since the three angles are congruent, the two triangles are similar.

See also

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