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

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

In this chapter, we will discuss another congruence theorem, this time the Side-Angle-Side theorem. The angle is called the included angle.

The Side-Angle-Side congruence theorem[edit]

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 are congruent and their other angles and sides are equal too. Success!

Proof[edit]

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 learnt to copy a line or a segment but we will move the triangle as whole.

  1. \text{Superpose } \triangle ABC \text{ on } \triangle DEF \text{ such that } A \text { is placed on } D \text { and } \overline {AB} \text{ is placed on } \overline {DE}
  2. \because \overline {AB} = \overline {DE} \text{ (given)}
  3. \therefore B \text{ coincides with } E
  4. \because \angle CAB = \angle FDE \text{ (given)}
  5. \therefore \overline {CA} = \overline {FD}
  6. \because \overline {CA} = \overline {DF} \text{ (given)}
  7. \therefore C \text{ coincides with } F
  8. \therefore \overline {CB} = \overline {EF}
  9. \therefore \triangle ABC = \triangle DEF
  10. \therefore \triangle ABC \cong \triangle DEF