Congruence Shortcuts Notes

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  • 1.
    • Two triangles are congruent if one can be placed on top of the other for a perfect match (they have the same size and shape).
    • In the figure, is congruent to In symbols:
    • Just as with similar triangles, it is important to get the letters in the correct order. For example, since A and D come first, we are saying that when the triangles are made to coincide, A and D will coincide.
    Definition of Congruent Triangles A B C D E F
  • 2. CPCTC
    • Corresponding parts of congruent triangles are congruent (CPCTC).
    • What this means is that if then:
    • Other corresponding “parts” (like medians) are also congruent.
    A B C D E F
  • 3.
    • To prove that two triangles are congruent it is only necessary to show that some corresponding parts are congruent.
    • For example, suppose that in and in that
    • Then intuition tells us that the remaining sides must be congruent, and…
    • The triangles themselves must be congruent.
    A D C B F E Proving Triangles Congruent
  • 4. SAS
    • In two triangles, if one pair of sides are congruent, another pair of sides are congruent, and the pair of angles in between the pairs of congruent sides are congruent, then the triangles are congruent.
    • For example, in the figure, if the corresponding parts are congruent as marked, then
    • We cite “Side-Angle-Side (SAS)” as the reason these triangles are congruent.
    A B C D E F
  • 5.
    • In two triangles, if all three pairs of corresponding sides are congruent then the triangles are congruent.
    • For example, in the figure, if the corresponding sides are congruent as marked, then
    • We cite “side-side-side (SSS)” as the reason why these triangles are congruent.
    SSS A B C D E F
  • 6. ASA
    • In two triangles, if one pair of angles are congruent, another pair of angles are congruent, and the pair of sides in between the pairs of congruent angles are congruent, then the triangles are congruent.
    • For example, in the figure, if the corresponding parts are congruent as marked, then
    • We cite “angle-side-angle (ASA)” as the reason the triangles are congruent.
    A B C D E F
  • 7. AAS
    • In two triangles, if one pair of angles are congruent, another pair of angles are congruent, and a pair of sides not between the two angles are congruent, then the triangles are congruent.
    • For example, in the figure, if the corresponding parts are congruent as marked, then
    • We cite “angle-angle-side (AAS)” as the reason the triangles are congruent.
    A B C D E F