4. Proving Triangle Similarity by AA
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The triangles are similar.
Let's consider two triangles with two pairs of corresponding congruent angles.
We want to establish a relation between the triangles above. To do so, we will start by performing a dilation on â–³ ABC by a scale factor of k= DFAC, and using vertex A as the center of dilation. Let â–³ AB'C' be the image of this dilation.
LHS * AC=RHS* AC
Cancel out common factors
Simplify quotient and product
By definition of congruent sides, we have that AC'≅ DF. Next, we will pay close attention to the angles. Because corresponding angles of similar triangles are congruent, we know that ∠C' ≅ ∠C. We already knew that ∠C ≅ ∠F. By the Transitive Property of Congruence, we know that ∠C' ≅ ∠F. ∠C' ≅ ∠C ∠C ≅ ∠F ⇒ ∠C' ≅ ∠F Also, by the Reflexive Property of Congruence, we can say that ∠A ≅ ∠A. Next, let's draw △ AB'C' and △ DEF and list the congruent parts between them.
By the Angle-Side-Angle Congruence Theorem, we can state that △ AB'C' ≅ △ DEF. Therefore, there is a composition of rigid motions that maps △ AB'C' to △ DEF.
This means that we can map â–³ ABC onto â–³ DEF by performing a dilation followed by a composition of rigid motions. Consequently, we can map â–³ ABC to â–³ DEF by applying a similarity transformation, which means that â–³ ABC ~ â–³ DEF.
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If two triangles have two pairs of corresponding angles that are congruent, then the triangles are similar. |