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Side Angle Side Theorem (Euclid Book 1, Proposition 4)

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In this example we will investigate the situation where two triangles ABC and DEF have two sides and the adjoining angle congruent.  In such a setting Euclid's side-angle-side theorem says that the triangles themselves will be congruent.  To start our investigation, construct a triangle ABC in the Canvas.

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Next, we will construct a triangle DEF with sides DE congruent to AB and DF congruent to AC. Also, our triangle will have angle EDF congruent to angle BAC.

First, create a point D somewhere away from triangle ABC and draw a ray from  D. Select D and then the segment AB. Use the circle tool in the Construct Panel to construct a circle at D of radius AB.  Then, find the intersection point E of the circle and the ray.

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Hide the circle, ray, and the ray point and additional intersection point.  Then, connect D and E with a segment. At this point segment DE will be congruent to segment AB.   Now we will construct an angle on DE that is congruent to BAC. Select D and choose "Center" from the "Mark" menu in the Transform Panel.  Then, select points B,A, and C (in that order) and select "Angle" from the "Mark" menu.  We have now defined an angle of rotation about point D.  Select point E.  The rotation button in the Transform Panel should now be active. Click on the rotation button to create point Q as shown. Then, create a ray from D through Q. (Make sure the ray gets "attached" at Q).

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As in the construction of point E, construct a circle of radius AC at point D. (Select D and then segment AC and click on the circle button in the Construct Panel).  Find the intersection point F of this circle with ray DQ.

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Hide the circle, the ray, point Q and the other intersection point.  Then, connect D to F and F to E completing the triangle DEF.

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At this point, we have two triangles that satisfy the assumptions of the side-angle-side theorem. Sides AB and DE are congruent, sides AC and DF are congruent, and angle BAC is congruent to angle EDF.  Let's look at what happens to the third side of these triangle and to the other two angles.

Select segments EF and BC and choose "Distance" under the Measure menu.  Then, select points F,E, and D (in that order) and choose "Angle" under the Measure menu. Also, measure the angles for CBA, DFE, and ACB.

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Note that sides EF and BC do appear congruent. Move points A, B, C, D around and check whether this congruence persists. We also note that all angles appear to be congruent in pairs. Moving points around appears not to alter this relationship.