Geometry for Elementary School/The Side-Angle-Side congruence theorem

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Geometry for Elementary School
Why the constructions are not correct? The Side-Angle-Side congruence theorem Some impossible constructions


In this chapter, we will discuss another congruence theorem, this time the Side-Angle-Side theorem. The theorem appears as Based on Book I, prop 4 at the Elements.

[edit] The Side-Angle-Side congruence theorem

Given two triangles \triangle ABC and \triangle DEF such that their sides are equal, hence:

  1. The side \overline {AB} equals \overline {DE} .
  2. The side \overline {CA} equals \overline {DF} .
  3. The angle \angle CAB equals \angle FDE (These are the angles between the sides).

Then the triangles congruent and their other angles and side are equal too. Success!

[edit] Proof

We will use the method of superposition – we will move one triangle to the other one and we will show that they coincide. We won’t use the construction we learned to copy a line or a segment but we will move the triangle as whole.

  1. Superpose \triangle ABC on \triangle DEF such that A is place on D and \overline {AB} is placed on \overline {DE} .
  2. It is given that \overline {AB} equals \overline {DE} .
  3. Hence, B coincides with E.
  4. It is given that the angle \angle CAB equals \angle FDE .
  5. Hence, \overline {CA} is placed on \overline {DF} .
  6. it is given that \overline {CA} equals \overline {DF} .
  7. Hence, C coincides with F.
  8. Therefore, \overline {CB} coincides with \overline {EF} .
  9. The triangles \triangle ABC and \triangle DEF coincide.
  10. The triangles \triangle ABC and \triangle DEF congruent.
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