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Difference between revisions of "AA similarity"

(Created page with "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 <math>\angle ...")
 
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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.
  
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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>.
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<math>\angle A + \angle B + \angle C = 180</math> (Sum of all angles in a triangle is <math>180</math>)
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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> (Sum of all angles in a triangle is <math>180</math>)
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<math>\angle D + \angle E + \angle F = 180</math>.
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<math>\angle A  + \angle B + \angle C=\angle D + \angle E + \angle F</math>
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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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<math>\angle D + \angle E + \angle C = \angle D + \angle E + \angle F</math> (since <math>\angle A = \angle D</math> and <math>\angle B = \angle E</math>)
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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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<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.
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==See also==
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* [[Similarity (geometry)]]
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* [[SAS similarity]]
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* [[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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