Proving Triangles are Congruent ASA and AAS

Similar Triangles:

When two triangles have the same shape but different sizes they are referred to as similar triangles. Symbolically, it can be written as, "~".

Congruent Triangles:

  When two triangles have the same shape and same size they are referred to as congruent triangles. They are also referred to as the special type of similar triangles.

Two triangles are said to congruent if,

• All 6 pairs of corresponding angles and sides are congruent.
• All 3 pairs of corresponding sides are the same.

Angle Side Angle (ASA) Postulate:

  If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then theses two triangles are congruent.

What is an included side?

An included side means the side included between the angles.

Consider the following example:

Prove that, ΔDEF is congruent to ΔGHI i.e.

ΔDEF = ΔGHI

To Prove: ΔDEF = ΔGHI

Proof:

• Two angles and the included side are congruent.
  • FDE = IGH (angle)
  • DE = GH (side)
  • DFE = GIH (angle)
• Hence, by the Angle Side Angle postulate (ASA), the triangles are congruent.

Angle Angle Side (AAS) Postulate:

If two angles and the non-included side one triangle are congruent to two angles and the non - included angle of another triangle, then these two triangles are congruent.

Consider the following example:

Prove that ΔDEF is congruent to ΔGHI i.e.

ΔDEF = ΔGHI

To Prove: ΔDEF = ΔGHI

Proof:

• Two angles and a non-included side are congruent.
• FDE = IGH (angle)
• DFE = GIH (angle)
• DE = GH (side)
• Hence, by the Angle Angle Side postulate (AAS), the triangles are congruent.