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2.4 – Congruence Postulates

Objectives

  • Discover the congruence shortcuts (SSS, SAS, and ASA postulates).
  • Use the AAS theorem to determine whether two triangles are congruent.
  • Experiment with SSA and AAA and discover why they do not work as congruence shortcuts.

 

Key Terms

  • Included Angle – An angle between two sides of a triangle
  • AAS Theorem – AAS stands for “angle-angle-side”
    • A theorem stating that if two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another, then the triangles are congruent.
    • This is a THEOREM because we can prove it using the “Third Angle Theorem”
  • ASA Postulate – ASA stands for “angle-side-angle”
    • A postulate stating that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.
  • SAS Postulate – SAS stands for “side-angle-side”
    • A postulate stating that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.
  • SSS Postulate – SSS stands for “side-side-side”
    • A postulate stating that if the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
    • Triangles with three pairs of congruent sides will always be congruent.
    • You can prove the SSS Postulate by using the other congruence postulates; so, it can be called a Theorem.
  • Triangle Congruence Postulate – A statement that proves triangles are congruent without requiring that the measures of all six pairs of corresponding parts be known.
  • Third Angle Theorem – If two angles of one triangle are congruent to two angles of another triangle, the third angles are also congruent.
    • You can combine ASA and AAS to make one general rule:  If two angles and any side of one triangle are congruent to the corresponding two angles and side of another triangle, the two triangles are congruent.

 

Notes

Review
  • What’s the difference between a postulate and a theorem?
    • A postulate (axiom) is a statement that is assumed to be true without proof.
    • A theorem is a statement that has been proven.
Congruence
  • If it is possible, it could always be true.
  • If it’s not possible, it’s not always true.

  • To determine congruence, you trace a triangle clockwise or counter-clockwise, looking for sides and angles.
  • You can only skip ONE, either a side OR an angle, when you trace the triangle in either direction.
    • SSA (Side-Side-Angle) or A-S-S does NOT guarantee (or prove) congruence!
      • Remember: ASS means donkey, not congruence!
    • AAA (Angle-Angle-Angle) does NOT guarantee (or prove) congruence!
      • Remember AAA tows your car, not your triangles!
    • See below (the triangles have the same angle measures, but one is larger than the other (not congruent)!

Geo A 2.04 - AAA No Congruence Guarantee

  • Proof by Contradiction (Indirect Proof): Side-Side-Side
    • We want to prove: If three pairs of segments are congruent, they always make two congruent triangles.
    • So we are proving: If three pairs of segments are congruent, it is impossible to make two triangles that are not congruent.
  • Proof by Contradiction (Indirect Proof): Side-Angle-Side
    • We want to prove: If two pairs of segments and their included angles are congruent, they always make two congruent triangles.
    • So we are proving: If two pairs of segments and their included angles are congruent, it is impossible to make two triangles that are not congruent.
  • Proof by Contradiction (Indirect Proof): Angle-Side-Angle
    • We want to prove: If two pairs of angles and their included sides are congruent, they always make two congruent triangles.
    • So we are proving: If two pairs of angles and their included sides are congruent, it is impossible to make two triangles that are not congruent.

Geo A 2.04 Proof


Geo A 2.04 - Congruence Postulates

Geo A 2.04 - Congruence Postulates Theorems

Examples
  • Example of ASA

Geo A 2.04 - ASA Example

Answer: \overline{AB} \cong \overline{XY}

  • Example of SAS

Geo A 2.04 - SAS Example

Answer: SAS

  • Example of AAS
    • True or False?

Geo A 2.04 - AAS Example

Answer: True by ASA

90^{\circ} (angle)-40^{\circ} (angle)-12 (side)

Note: Ignore the fact that they are NOT drawn to scale and are reflected.

 

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