If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
Based on the diagram above, the theorem can be written as follows.
⎩⎪⎪⎨⎪⎪⎧∠A≅∠DAB≅DE∠B≅∠E⇒△ABC≅△DEF
Proof
Angle-Side-Angle Congruent Theorem
This proof will be developed based on the given diagram, but it is valid for any pair of triangles.
The goal of the proof is to find a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways. One of the ways will be shown here.
1
Translate △DEF So That Two Corresponding Vertices Match
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Translate△DEF so that D is mapped onto A. If this translation maps △DEF onto △ABC, the proof is complete.
Since the image of the translation does not match △ABC, at least one more transformation is needed.
2
Rotate △AE′F′ So That Two Corresponding Sides Match
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Rotate△AE′F′ counterclockwise about A so that a pair of corresponding sides match. If the image of this transformation is △ABC, the proof is complete. Note that this rotation maps E′ onto B. Therefore, the rotation maps AE′ onto AB.
As before, the image does not match △ABC. Therefore, a third rigid motion is required.
3
Reflect △ABF′′ So That All Corresponding Sides Match
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Reflect△ABF′′ across AB. Because reflections preserve angles, AF′′ and BF′′ are mapped onto AC and BC, respectively. Then, the point of intersection of the original raysF′′ is mapped onto the point of intersection of the image rays C.
This time the image matches △ABC.
Consequently, after a sequence of rigid motions △DEF can be mapped onto △ABC. This means that △DEF and △ABC are congruent triangles.
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