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